As shown in the figure, in parallelogram $$\square A B C D$$, if $$B A = B D$$, $$A E \bot B D$$, $$\angle C = 70^\circ$$, then $$\angle D A E =$$         degrees.

Answer: $$20$$


As shown in the figure, $$BD$$ is the median of $$\triangle ABC$$, $$AB = 6$$, $$BC = 4$$, the difference in the perimeters of $$\triangle ABD$$ and $$\triangle BCD$$ is      .

Answer: 2


In rectangle $$A B C D$$, $$A B = 3$$, $$A D = 2 \sqrt{13}$$, point M is on side $$B C$$, connect $$A M$$, fold $$\triangle A B M$$ along $$A M$$ to obtain $$\triangle A B^{'} M$$, $$B^{'} M$$ intersects $$A D$$ at point N, if point N is the midpoint of $$A D$$, then the length of $$B M$$ is                   .

Answer: $$2 + \sqrt{13}$$/$$\sqrt{13} + 2$$


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, and point D is any point on side $$B C$$ other than B and C. $$D E \bot A B$$ at point E, and $$D F \bot A C$$ at point F. If the altitude from side $$B C$$, $$A M = 10$$, then $$D E + D F =$$           .

Answer: 10


As shown in the figure, △ABC ≌ △DBE, the perimeter of △ABC is 30, AB = 9, BE = 8, then the length of AC is  .

Answer: 13


As shown in the figure, $$AD$$ is the median of $$\triangle ABC$$. If $$S_{\triangle ABC} = 2$$, then $$S_{\triangle ACD} =$$        .

Answer: $$1$$


As shown in the figure, in $$\text{Rt} \triangle \text{ABC}$$, $$\angle A C B = 90 \circ$$, $$\angle A B C = 60 \circ$$, $$B C = 2$$. If a semicircle is drawn with $$A B$$ as the diameter, and an arc is drawn with point $$B$$ as the center and $$B C$$ as the radius, intersecting $$A B$$ at point $$D$$, then the area of the shaded region is                     . (Express the result in terms of $$\pi$$)

Answer: $$\frac{2}{3}$$π+$$\sqrt{3}$$


As shown in the figure, $$AF \parallel CD$$, $$BD$$ bisects $$\angle EBF$$, and $$BC \bot BD$$. The following conclusions: ① $$BC$$ bisects $$\angle ABE$$; ② $$AC \parallel BE$$; ③ $$\angle CBE + \angle D = 90^\circ$$; ④ $$\angle DEB = 2 \angle BCD$$. Among these, the correct conclusions are        (only fill in the serial numbers).

Answer: ①③④


As shown in the figure, in $$\bigodot O$$, $$A$$ is a point on the major arc $$BC$$, $$\angle BAC = \alpha$$, connect $$BO$$ and $$CO$$, extend $$BO$$ to intersect $$AC$$ at point $$D$$, then the angle in the figure that measures $$2\alpha$$ is      .

Answer: $$\angle B O C$$


As shown in the figure, a triangular cardboard with an area of 10, points D, E, F are the midpoints of segments AF, BD, CE respectively, then the area of the shaded region is      .

Answer: $$\frac{10}{7}$$/$$1 \frac{3}{7}$$


As shown in the figure, if $$\triangle A B C \approx \triangle E B D$$, and $$B D = 4, A B = 8$$, then the area of the shaded region $$S_{\triangle A C E} =$$        .

Answer: 16


As shown in the figure, in the right triangle $$Rt \triangle ABC$$, $$\angle ACB = 90^\circ$$, $$D$$ is the midpoint of $$AB$$, $$CD = 5$$, then $$AB =$$      .

Answer: 10


As shown in the figure, the diagonals $$AC, BD$$ of rhombus $$ABCD$$ intersect at point $$O$$, $$AE \bot BC$$, with foot at $$E$$. If $$AC = 6, BD = 8$$, then the length of $$AE$$ is      .

Answer: $$\frac{24}{5}$$


As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C, B D$$ intersect at point O, $$\angle A O D = 116 \circ$$, the measure of $$\angle A C D$$ is         .

Answer: $$58 \circ$$


As shown in the figure, in right triangles Rt△BAC and Rt△BDC, ∠BAC = ∠BDC = 90°, O is the midpoint of BC, and AO and DO are connected. If AO = 3, then the length of DO is     .

Answer: 3


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, diagonals $$A C$$, $$B D$$ intersect at point O, and $$A E$$ perpendicularly bisects $$O B$$ at point E. Then the length of $$A D$$ is            .

Answer: $$3 \sqrt{3}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, $$D$$, $$E$$, $$F$$ are the midpoints of $$A B$$, $$B C$$, $$C A$$ respectively. If $$D E = 3$$, then $$B F =$$        .

Answer: 3


As shown in the figure, in the isosceles triangle $$\triangle ABC$$, $$AB = AC = 8$$, $$\angle BAC = 90^\circ$$, the circle $$\bigodot O$$ with diameter $$AB$$ intersects $$BC$$ at point $$D$$. Connect $$OD$$ and $$AD$$. The area of the shaded region in the figure is         .

Answer: $$4 \pi - 8$$


As shown in the figure, in $$\triangle A B C$$, angles $$\angle B$$ and $$\angle C$$ are folded as shown in the figure, with points B and C both landing at a point G on side $$B C$$, and segments $$M N$$, $$E F$$ are the fold lines. If $$\angle A = 94 \circ$$, then $$\angle M G E =$$      .

Answer: $$94 \circ$$


As shown in the figure, given AB ∥ DE, ∠ABC = 75°, ∠CDE = 150°, find the measure of ∠BCD.

Answer: 45°


As shown in the figure, in right triangle $$\triangle$$ABC, ∠B=90°, ED is the perpendicular bisector of AC, intersecting AC at point D and BC at point E. Given that ∠BAE=10°, the measure of ∠C is             °.

Answer: 40°


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle A = 50 \circ$$. It is folded such that point A lands at point $$A^{'}$$ on side $$C B$$, with the crease being $$C D$$. Find the measure of $$\angle A^{'} D B$$.

Answer: $$10 \circ$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A D$$ is the median of $$\triangle A B C$$, points E and F are the midpoints of $$A D$$ and $$A C$$ respectively, connect $$E F$$, if $$E F = 3$$, then the length of $$A D$$ is        .

Answer: $$6$$


In the ancient Chinese mathematical text "The Nine Chapters on the Mathematical Art," there is the following problem: "A bamboo stick is 1 zhang tall. After it breaks, the top of the bamboo falls 3 feet away from the base of the bamboo. How high above the ground is the break point?" (Note: 1 zhang = 10 feet). As shown in the figure, according to the problem, let the top of the broken bamboo fall at point A, the base of the bamboo be point B, and the break point be point C. The length of BC, the height of the break point from the ground, is              feet.

Answer: 4.55


As shown in the figure, in the right triangle paper $$ABC$$, $$\angle ACB = 90^\circ$$, $$AC = 6$$, $$BC = 8$$. Fold the paper so that point B coincides with point A, then unfold to obtain the crease $$DE$$. The length of $$CD$$ is      .

Answer: $$\frac{7}{4}$$


As shown in the figure, the side length of square $$A B C D$$ is 2. $$E$$ is a moving point that does not coincide with point $$D$$; construct square $$D E F G$$ with $$D E$$ as one side. Let $$D E = d_{1}$$, and the distances from points $$F$$ and $$G$$ to point $$C$$ are $$d_{2}$$ and $$d_{3}$$, respectively. Then the minimum value of $$d_{1} + d_{2} + d_{3}$$ is        .

Answer: $$2 \sqrt{2}$$


As shown in the figure, to cover a staircase that is 5 m high and 13 m long with red carpet, at least      m of red carpet is needed.

Answer: 17


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 6$$, $$B C = 8$$, point $$M$$ is the midpoint of $$A B$$, find $$C M =$$       .

Answer: 5


As shown in the figure, $$\triangle O A_{1} A_{2}$$ is an isosceles right triangle, $$O A_{1} = 1$$, and an isosceles right triangle $$\text{Rt} \triangle O A_{2} A_{3}$$ is constructed with hypotenuse $$O A_{2}$$ as one leg; then an isosceles right triangle $$\text{Rt} \triangle O A_{3} A_{4}$$ is constructed with $$O A_{3}$$ as one leg, and so on. Following this pattern, a spiral pattern is obtained. The length of $$O A_{n}$$ is        .  (Expressed in terms of n)

Answer: $$\left(\sqrt{2}\right)^{n - 1}$$


As shown in the figure, $$BD$$ is the diagonal of rectangle $$ABCD$$, $$BE$$ bisects $$\angle ABD$$, if $$\angle BDC = 50^{\circ}$$, then $$\angle AEB =$$       °.

Answer: $$65$$


As shown in the figure, in parallelogram $$\square ABCD$$, $$AC$$ is a diagonal, $$\angle ACD = 90^\circ$$, E is the midpoint of $$BC$$, $$AF$$ bisects $$\angle BAC$$, and connect $$CF$$, $$EF$$. If $$CF \perp AF$$, $$AB = 5$$, $$BC = 13$$, then the length of $$EF$$ is       .

Answer: $$\frac{7}{2}$$/3.5


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$\angle B = 62 \circ$$, D and E are on $$A B$$ and $$A C$$ respectively. Folding $$\triangle A D E$$ along $$D E$$ gives $$\triangle F D E$$, and it satisfies $$E F \parallel A B$$. Then $$\angle 1 =$$        .

Answer: $$76 \circ$$


As shown in the figure, which is formed by combining a set of triangle rulers, the degree measure of ∠ABC in the figure is        .

Answer: 75°


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle B = 90 \circ$$, point $$E$$ is a moving point on side $$B C$$, $$D C \bot B C$$, connect $$A E$$, $$D E$$. $$D E$$ intersects $$A C$$ at point $$F$$, $$\angle D F C = 45 \circ$$, $$A C = 2 \sqrt{10}$$, $$C E = 3 \sqrt{2}$$, if $$B E = D C$$, then $$A E$$      .

Answer: $$\sqrt{10}$$


As shown in the figure, in $$\triangle ABC$$, $$AC = BC$$, $$\angle C = 30^\circ$$, $$BD$$ bisects $$\angle ABC$$ and intersects $$AC$$ at $$D$$, points $$E$$ and $$F$$ are on $$AB$$ and $$BC$$ respectively. When $$\triangle BEF$$ is folded along $$EF$$, point $$B$$ coincides exactly with point $$D$$. If the perimeter of quadrilateral $$BEDF$$ is 16, then the length of $$CD$$ is           .

Answer: $$2 \sqrt{6} + 2 \sqrt{2}$$/$$2 \sqrt{2} + 2 \sqrt{6}$$


In Rt△ACB, ∠ACB=90°, point D is on side AB, connect CD, fold △ADC along the straight line CD, point A happens to fall on point E on side BC, if AC=3, BE=1, then the length of DE is     .

Answer: $$\frac{15}{7}$$


As shown in the figure, this is a schematic diagram of the "string figure" used by the ancient Chinese mathematician Zhao Shuang to prove the Pythagorean theorem. It consists of four congruent right triangles and a small square $$EFGH$$, exactly forming a large square $$ABCD$$. Connect $$EG$$ and extend it to intersect $$BC$$ at point $$M$$. If $$AB = 5$$ and $$EF = 1$$, then the length of $$GM$$ is        .

Answer: $$\frac{9 \sqrt{2}}{7}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = 10, t a n A = 2, B E \bot A C$$ at point E, and D is a moving point on segment $$B E$$, then the minimum value of $$\sqrt{5} C D + B D$$ is          .

Answer: 20


As shown in the figure, the square $$ABCD$$ has an area of 16, and its diagonals $$AC$$, $$BD$$ intersect at point $$O$$. Points $$E$$ and $$F$$ move along sides $$AB$$ and $$BC$$ respectively, with $$\angle EOF = 90^\circ$$, and $$OG$$ bisects $$\angle EOF$$, intersecting side $$BC$$ at point $$G$$. The following conclusions are given:
① $$OE = OF$$;
② The area of quadrilateral $$OEBF$$ remains constant at 4;
③ $$BG^{2} + CF^{2} = GF^{2}$$;
④ The minimum value of $$EF$$ is $$2\sqrt{2}$$.
Which of the above statements are correct? (Fill in all the correct statement numbers)

Answer: ①②③④


As shown in the figure, in the circle $$\bigodot O$$ with radius 3, B is the midpoint of the minor arc AC. Connect AB and extend it to D such that $$BD = AB$$. Connect AC, BC, and CD. If $$AB = 2$$, then CD equals      .

Answer: $$\frac{4}{3}$$


As shown in the figure, in the isosceles right triangle $$ABC$$, $$AB = BC = 4\sqrt{2}$$, point $$P$$ lies on the circle with diameter $$AC$$, and $$M$$ is the midpoint of $$PB$$. When point $$P$$ moves along the circle from point $$A$$ through one full revolution, the minimum value of the length of $$CM$$ is      .

Answer: $$2 \sqrt{5} - 2$$


As shown in the figure, in square ABCD, the sides of ∠EAF intersect the extensions of CB and DC at points E and F respectively, and ∠EAF = 45°. If BE = 1 and DF = 7, then EF =  .

Answer: 6


In the rhombus $$A B C D$$, $$A B = 4$$, $$\angle B = 2 \angle A$$, points E and F are the midpoints of $$A D$$ and $$A B$$ respectively. A moving point P starts from B and moves clockwise to point C. When $$\triangle P E F$$ is a right triangle, the length of $$B P$$ is      .

Answer: 3 or $$\sqrt{13}$$ or $$2 \sqrt{3}$$


As shown in the figure, point E is a moving point on the diagonal AC of rectangle ABCD. From point E, draw EF ⊥ BC, intersecting at point F. Given AB = 3 and BC = 6, if there is a point G on AB such that triangle EFG is an isosceles right triangle, then the length of AG is ______.

Answer: 3 or 1 or $$\frac{9}{5}$$


As shown in the figure, in square $ABCD$ with side length $4$, point $E$ is the midpoint of side $BC$, and point $F$ is a moving point on side $AB$. An equilateral triangle $FEG$ is constructed with $EF$ as one side, located above and to the right of $EF$. When $CG$ is minimized, the perimeter of $\triangle ECG$ is      .

Answer: $$5 + \sqrt{7}$$/$$\sqrt{7} + 5$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ, \angle B = 30 \circ$$.D is a moving point on side $$B C$$, and connect $$A D$$.When $$A D + \frac{1}{2} B D$$ takes its minimum value, the value of $$\frac{B D}{B C}$$ is        .

Answer: $$\frac{2}{3}$$


In $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C$$, draw a line $$l$$ through point $$C$$ parallel to $$A B$$, and let point $$P$$ be a moving point on line $$l$$ different from point $$C$$. Connect $$A P$$, and from point $$P$$ draw a perpendicular to $$A P$$ intersecting line $$B C$$ at point $$D$$. If $$A C = 7 \sqrt{2}$$, $$A P = 25$$, then the length of segment $$B D$$ is       .

Answer: $$17 \sqrt{2}$$ or $$31 \sqrt{2}$$


As shown in the figure, the diameter AB of ⊙O is 2, and C is a fixed point on ⊙O, with ∠ABC = 30°. A moving point P starts from A and moves along the semicircular arc $$\hat{AB}$$ toward point B (point P and point C lie on opposite sides of diameter AB). The motion stops when P reaches point B. During the motion, from point C, draw CD perpendicular to CP, intersecting the extension of PB at point D, and connect AD. The maximum value of segment AD is                   .

Answer: $$\sqrt{7} + \sqrt{3}$$/$$\sqrt{3} + \sqrt{7}$$


As shown in the figure, AP and BP bisect the interior angle ∠CAB and the exterior angle ∠CBD of △ABC, respectively. Connect CP. If ∠ACP = 130°, then ∠APB =   .

Answer: $$40 \circ$$


As shown in the figure, in square $$ABCD$$, $$AB = 7$$, F is a moving point on side $$CD$$. Rotate $$\triangle ADF$$ clockwise about point A by $$90^\circ$$ to obtain $$\triangle ABE$$, and reflect $$\triangle ADF$$ across line AF to obtain $$\triangle AGF$$. Let EF and BD intersect at point H, and connect GH. Then the maximum area of $$\triangle EGH$$ is       .

Answer: $$\frac{49}{16}$$


As shown in the figure, the equilateral triangle $$\triangle A B C$$ has side length 4, and the radius of $$\bigodot C$$ is 2. Point P is a moving point on $$A B$$, and from point P, a tangent $$P Q$$ is drawn to $$\bigodot C$$, with Q being the point of tangency. Then the minimum value of $$P Q$$ is                .

Answer: $$2 \sqrt{2}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$A D = 8$$, $$E$$ is a moving point on $$B C$$ (not coinciding with points $$B$$, $$C$$). Connect $$A E$$, and from point $$D$$, draw $$D F \bot A E$$, with foot of perpendicular at $$F$$. Then the minimum length of segment $$B F$$ is     .

Answer: $$2 \sqrt{13} - 4$$


As shown in the figure, in △ABC, ∠BAC = ∠BCA = 44°, M is a point inside △ABC; and ∠MCA = 30°, ∠MAC = 16°, then the measure of ∠BMC is    .

Answer: 150°


As shown in the figure, in quadrilateral $$ABCD$$, $$AB \parallel CD$$, $$\angle B = \angle C = 90^\circ$$, $$\triangle ADE$$ is an equilateral triangle, and point $$E$$ lies on $$BC$$. If $$AB = 6$$, $$CD = 9$$, the area of $$\triangle ADE$$ is        .

Answer: $$21 \sqrt{3}$$


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, $$A B = 6$$, $$P$$ is a moving point on $$A B$$, $$P D \bot B C$$, $$P E \bot A C$$, then the minimum value of $$D E$$ is        .

Answer: $$\frac{9}{2}$$


As shown in the figure, in the rhombus $$A B C D$$, point $$E$$ is a point on side $$A D$$, connect $$B E$$, rotate segment $$E B$$ counterclockwise around point $$E$$ by $$120^{\circ}$$ to obtain $$E F$$, $$E F$$ intersects $$C D$$ at point $$H$$, connect $$B F$$, which intersects $$C D$$ at point $$G$$. Given $$\angle A = 120^{\circ}$$, $$A B = 3$$, $$A E = 1$$, then $$B E =$$    , $$\frac{F G}{B G}$$=      .

Answer: $$\sqrt{13}$$     $$\frac{2}{3}$$


As shown in the figure, point O is the center of square $$A B C D$$, with $$A B = 3 \sqrt{2}$$. In $$R t \triangle B E F$$, $$\angle B E F = 90^\circ$$, and $$E F$$ passes through point D; $$B E$$ and $$B F$$ intersect $$A D$$ and $$C D$$ at points G and M, respectively. Connect $$O E$$, $$O M$$, and $$E M$$. If $$B G = D F$$ and $$\tan \angle A B G = \frac{1}{3}$$, then the perimeter of $$\triangle O E M$$ is           .

Answer: $$3 + 3 \sqrt{5}$$


As shown in the figure, in △ABC, D is the midpoint of AC, the angle bisector AE of △ABC intersects BD at point F. If BF:FD = 3:1, and AB + BE = 3$$\sqrt{3}$$, then the perimeter of △ABC is     .

Answer: $$5 \sqrt{3}$$


In triangle $$\triangle A B C$$, $$A C = 8$$, $$A B = \sqrt{41}$$, and the height from $$A$$ to side $$B C$$, denoted as $$A G = 5$$. Point $$D$$ is a moving point on segment $$A C$$. On $$B C$$, take $$C E = A D$$, and connect $$A E$$ and $$B D$$. Then the minimum value of $$A E + B D$$ is      .

Answer: 13


As shown in the figure, in rectangle $$A B C D$$, $$A B$$ = 6, $$B C$$ = 8, points $$M$$ and $$N$$ are the midpoints of sides $$A D$$ and $$B C$$, respectively. At a certain moment, point $$E$$ starts from point $$M$$ and moves uniformly toward point $$A$$ along the direction $$M A$$ at a speed of 2 units per second; simultaneously, point $$F$$ starts from point $$N$$ and moves uniformly toward point $$C$$ along the direction $$N C$$ at a speed of 1 unit per second. When one of the points reaches a vertex of the rectangle, both points stop moving. Connect $$E F$$, and from point $$B$$, draw a perpendicular to $$E F$$, with foot of the perpendicular at $$H$$. During this motion, the length of the path traced by point $$H$$ is     .

Answer: $$\frac{\sqrt{5}}{2} \pi$$/$$\frac{\sqrt{5} \pi}{2}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$\angle B = 30^{\circ}$$, $$A C = 2$$, point $$D$$ is the midpoint of $$B C$$, and point $$E$$ is a moving point on side $$A B$$. Fold $$\triangle B D E$$ along the line $$D E$$ to the position $$\triangle B^{'} D E$$, with $$B^{'} D$$ intersecting $$A B$$ at point $$F$$, and connect $$A B^{'}$$.

(1) The minimum value of $$A B^{'}$$ is        ; (2) If $$\triangle A B^{'} F$$ is a right triangle, then the length of $$B E$$ is        .

Answer: $$\sqrt{7} - \sqrt{3}$$     1 or $$\frac{6}{5}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$A D \parallel B C$$, a circle passing through points A, C, and D intersects $$B C$$ at point E. Connect $$A E$$. It is given that $$\overset{⌢}{A E} = \overset{⌢}{C E}$$, $$\angle B A E = \angle B E A$$. If $$B E = \frac{5}{6} C E = \frac{25}{6}$$, then the radius of the circle is        .

Answer: $$\frac{5 \sqrt{5}}{2}$$/$$\frac{5}{2} \sqrt{5}$$


As shown in the figure, $$\bigodot O$$ is the circumcircle of $$\triangle A B C$$, $$A B = A C$$, $$C D \bot A B$$ at point $$D$$, the extension of $$B O$$ intersects $$C D$$ at point $$E$$ and intersects $$\bigodot O$$ at point $$F$$.

(1) Let $$\angle B A C = \alpha$$, then express $$\angle D C F =$$      in terms of $$\alpha$$.
(2) If $$B C = 4 \sqrt{2}$$, $$B E = 4$$, then $$O E =$$      .

Answer: $$90 \circ - \frac{1}{2} \alpha$$     $$1$$


As shown in the figure, in $$\triangle A B C$$, $$B C = 6$$, $$E$$ and $$F$$ are the midpoints of $$A B$$ and $$A C$$ respectively, point $$P$$ moves along ray $$E F$$, $$B P$$ intersects $$C E$$ at $$D$$, the angle bisector of $$\angle C B P$$ intersects $$C E$$ at $$Q$$, when $$C Q = \frac{1}{2} C E$$, $$E P + B P =$$       .

Answer: $$12$$


As shown in the figure, in the rhombus $$A B C D$$, $$\angle A B C = 60 \circ$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, and $$P$$ is a moving point on diagonal $$B D$$, with $$P M \bot A B$$ at point $$M$$ and $$P N \bot A D$$ at point $$N$$. The following conclusions are given: ① $$\triangle A B C$$ is an equilateral triangle, ② $$O B = \sqrt{3} O A$$, ③ $$\angle M P N = 60 \circ$$, ④ $$P M + P N = \frac{1}{2} B D$$. Among these, the correct ones are         (fill in the serial numbers)

Answer: ①②③④


As shown in the figure, quadrilateral $$A B C D$$ is a rhombus, $$A B = 4$$, $$\angle A B C = 60 \circ$$, $$E$$ and $$F$$ are two moving points on $$B D$$, and $$B E + D F = E F$$, point $$M$$ is the midpoint of $$A D$$, connect $$C E$$ and $$M F$$, then the minimum value of $$C E + M F$$ is      .

Answer: $$2 \sqrt{3}$$


As shown in the figure, the angle bisectors BP and AP of ∠ABC and ∠EAC intersect at point P. PM ⊥ BE and PN ⊥ BF, with feet of perpendiculars at M and N, respectively. There are four conclusions:
① CP bisects ∠ACF; ② ∠BPC = $$\frac{1}{2}$$∠BAC; ③ ∠APC = 90° − $$\frac{1}{2}$$∠ABC; ④ S△APM + S△CPN > S△APC.
Among these, the correct conclusions are     . (fill in the conclusion numbers)

Answer: ①②③


As shown in the figure, $$\angle A O B = 45 \circ$$, there are two moving points C and D on sides $$O A$$ and $$O B$$ respectively. Connect $$C D$$, and construct an isosceles right triangle $$C D E$$ with $$C D$$ as a leg. When the length of $$C D$$ remains constant and equals $$2 \text{cm}$$, the maximum value of $$O E$$ is                    .

Answer: $$\sqrt{2} + \sqrt{10}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$A D = A B = B C$$, $$A C \bot B D$$ intersecting at point $$O$$, from vertex $$A$$ of quadrilateral $$A B C D$$, draw $$A F \bot A D$$, and $$A D = A F$$, segment $$A F$$ intersects $$O B$$ at point $$H$$ and intersects $$B C$$ at point $$E$$. If points $$D$$, $$C$$, $$F$$ are collinear, then the following statements: $$\textcircled{1}$$ quadrilateral $$A B C D$$ is a rhombus; $$\textcircled{2}$$ $$A B \cdot A E = A C \cdot B D$$; $$\textcircled{3}$$ $$\angle A H D = \angle C A D$$; $$\textcircled{4}$$ $$O H + O C = \frac{3}{4} B H$$, the correct ones are      .

Answer: $$\textcircled{1} \textcircled{3}$$


As shown in the figure, in quadrilateral ABCD, AD = AB, DC = BC, ∠DAB = 60°, ∠DCB = 120°, E is on AD, F is a point on the extension of AB, and DE = BF. If G is on AB and ∠ECG = 60°, then the quantitative relationship among DE, EG, and BG is     .

Answer: DE+BG=EG


As shown in the figure, rectangle $$A B C D$$ (opposite sides of a rectangle are equal, each angle is $$90 \circ$$), $$A B = 6 \text{cm}$$, $$A D = 2 \text{cm}$$, point $$P$$ and point $$Q$$ start simultaneously from points $$A$$ and $$C$$, respectively. Point $$P$$ moves toward endpoint $$B$$ at a speed of 2 cm/s, and point $$Q$$ moves toward $$D$$ at a speed of 1 cm/s. When one point reaches its endpoint, the other point also stops moving. Let the time of motion be $$t$$.

(1) When the distance between points $$P$$ and $$Q$$ is $$3 \text{cm}$$, $$t =$$          . (2) When $$t =$$       , $$\triangle P Q D$$ is a right triangle ($$t \neq 0$$).

Answer: $$t = \frac{6 - \sqrt{5}}{3}$$ or $$\frac{6 + \sqrt{5}}{3}$$     $$t = \frac{3 + \sqrt{3}}{3}$$ or $$\frac{3 - \sqrt{3}}{3}$$ or 2


As shown in the figure, in $$\triangle A B C$$, points $$D,  E,  F$$ are on sides $$A B,  B C,  A C$$ respectively, and $$A E,  B F,  C D$$ intersect at point $$G$$. If $$\frac{A G}{G E} + \frac{B G}{G F} + \frac{C G}{G D} = 2014$$, then the value of $$\frac{A G}{G E} \cdot \frac{B G}{G F} \cdot \frac{C G}{G D}$$ is            .

Answer: 2016


As shown in the figure, square $$ABCD$$, point $$E$$ is a moving point on ray $$AB$$, draw $$EF \parallel DB$$, intersecting line $$AD$$ at point $$F$$, connect $$DE$$, take the midpoint $$G$$ of $$DE$$, connect $$FG$$ and extend it to intersect line $$DB$$ at point $$H$$. If $$AB = 4$$, $$EB = 3$$, then the length of $$FH$$ is      .

Answer: $$\sqrt{5}$$ or $$\sqrt{149}$$/$$\sqrt{149}$$ or $$\sqrt{5}$$


As shown in the figure, the side length of square $$A B C D$$ is 8, point E is on $$D C$$, point F is on $$A C$$, $$\angle B F E = 90 \circ$$, and if $$C E = 2$$, then the length of $$A F$$ is      .

Answer: $$\text{3} \sqrt{\text{2}}$$


As shown in the figure, in △ABC, AB = AC, BD ⊥ AC at point D. Rotate segment AC about point C to obtain segment CE, such that point E lies exactly on the extension of AB. $$BE = \frac{1}{2} CD$$, and the area of △BCD is 8. Find the length of BC.

Answer: $$2 \sqrt{10}$$


As shown in the figure, in $$\triangle A B C$$, $$A B - A C = 4$$, $$B C = 7$$, $$B D$$ is perpendicular to the angle bisector $$A D$$ of $$\angle B A C$$ at point $$D$$, $$E$$ is the midpoint of $$A C$$, and $$B E$$ intersects $$A D$$ at $$F$$. Find the maximum value of the difference in areas of $$\triangle B D F$$ and $$\triangle A E F$$.

Answer: $$7$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$A D \bot B C$$ at point $$D$$, $$D E \bot A B$$ at point $$E$$. If $$\angle B = 30 \circ$$, $$A E = 1$$.

(1) The length of $$B E$$ is      ; (2) Take a point $$M$$ on one of the legs of $$\triangle A B C$$, when $$\triangle D E M$$ is an isosceles triangle, the length of $$B M$$ is      .

Answer: $$3$$     $$3 - \sqrt{3}$$ or $$\sqrt{21}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, point D is the midpoint of the hypotenuse $$A B$$, from point B, draw $$B E \bot C D$$ intersecting at F, intersecting $$A C$$ at point E, from point A, draw $$A G \parallel B C$$, intersecting the extension of $$B E$$ at point G. If $$A E = B C$$, then the value of $$\frac{D F}{C F}$$ is      .

Answer: $$\frac{\sqrt{5} + 1}{4}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, it is given that $$\angle A C B = 90^{\circ}$$, $$A C = 4$$, $$B C = 2 \sqrt{5}$$, points $$D$$ and $$E$$ are two moving points on sides $$A B$$ and $$A C$$ respectively, and $$A D = C E$$ always holds. Connect $$B E$$ and $$C D$$. When $$C D + B E$$ takes its minimum value, $$A E =$$           .

Answer: $$\frac{12}{5}$$


As shown in the figure, in quadrilateral ABCD, the angle bisector of $$\angle DAB$$ intersects the external angle bisector of $$\angle ABC$$ at point P, and $$\angle D + \angle C = 240^\circ$$. Then $$\angle P =$$      .

Answer: $$30 \circ$$


As shown in the figure, in the equilateral triangle $$\triangle A B C$$ with side length 4, D and E are the midpoints of $$A B$$ and $$B C$$ respectively. Connect $$D E$$, and let F be the midpoint of $$D E$$. Connect $$A F$$. Then the length of $$A F$$ is         .

Answer: $$\sqrt{7}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B =\text{60} \circ$$, rotate $$\triangle A B C$$ clockwise about point C to the position of $$\triangle C D E$$, let F be a point on CE, connect AF and extend it to intersect the extension of BC at point G. If CF bisects $$\angle A C G$$, $$\angle C A F = 2 \angle E A F$$, $$\angle A E D = 21 \circ$$, then $$\angle G$$ is      °.

Answer: $$34$$


As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, $$A B = 8$$, $$B C = 10$$, diagonal $$B D = 6$$, P is a moving point on $$A D$$, and Q is a fixed point on $$A B$$. The minimum value of $$3 D P + 5 P Q$$ is         .

Answer: 30


As shown in the figure, in quadrilateral ABCD, ∠C + ∠D = $$210 \circ$$, E and F are points on AD and BC respectively. Quadrilateral CDEF is folded along line EF to obtain quadrilateral $$C^{'} D^{'} E F$$, $$C^{'} F$$ intersects AD at point G. If △EFG has two equal angles, then ∠EFG =          .

Answer: $$40 \circ$$ or $$50 \circ$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$A D = 4$$, point $$E$$ is the midpoint of $$A D$$, point $$F$$ lies on line $$C D$$, point $$G$$ lies on segment $$E F$$, and $$\angle G D F = \angle D E F$$, point $$P$$ is a moving point on side $$B C$$, then the minimum value of $$P A + P G$$ is      .

Answer: $$3 \sqrt{5} - 1$$/$$- 1 + 3 \sqrt{5}$$


As shown in the figure, in $$\triangle A B C$$, $$B D$$ is the median to side $$A C$$, and $$A E$$ is the median to side $$B D$$ in $$\triangle A B D$$. If $$\angle C B D = 60 \circ$$, $$\angle A E B = 150 \circ$$, and $$B D = 4$$, then $$A B =$$          .

Answer: $$7$$


In △ABC, $$\angle A C B = 90 \circ, \angle B = 60 \circ,$$ $$A B = 4,$$ point D is a moving point on line BC, connect AD, construct an equilateral △ADE on the right side of line AD, connect CE, when the length of segment CE is minimized, the length of segment $$C D$$ is   .

Answer: $$3$$


As shown in the figure, the side length of square $$A B C D$$ is $$4$$, and points $$E$$ and $$F$$ are on sides $$B C$$ and $$C D$$ respectively, with $$C E = D F$$. Lines $$A F$$ and $$D E$$ intersect at point $$O$$, and $$B O = B A$$. Find the value of $$O C$$.

Answer: $$\frac{4 \sqrt{10}}{5}$$


As shown in the figure, given that the side length of square $$A B C D$$ is $$3$$, and $$E$$, $$F$$ are points on sides $$A B$$ and $$B C$$ respectively, with $$\angle E D F = 45^\circ$$. Rotate $$\triangle D A E$$ counterclockwise about point $$D$$ by $$90^\circ$$ to obtain $$\triangle D C M$$. If $$A E = 1$$, then the length of $$F M$$ is  .

Answer: $$2.5$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, $$AB = 2$$, points C and D are on the same side of $$AB$$, and $$AD \bot AB$$, $$BC \bot AB$$, $$AD = 1$$, $$BC = 3$$. Point P is a moving point on $$\bigodot O$$. Find the minimum value of $$\frac{\sqrt{2}}{2} PD + PC$$ .

Answer: $$\frac{\sqrt{34}}{2}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$B E = 6$$, $$\angle B = 60 \circ$$, point $$P$$ is a moving point on segment $$A C$$, and point $$F$$ is a moving point on segment $$A B$$. When $$E P + F P$$ is minimized, connect $$E F = 6$$, then the length of $$A B$$ is         .

Answer: $$\text{18}$$


As shown in the figure, in rectangle $$ABCD$$, $$BC = 2AB$$, point P is a moving point on side $$AD$$, and an equilateral triangle $$\triangle BPP'$$ is constructed to the right with $$BP$$ as its side. Connect $$CP'$$. When point $$P'$$ falls on side $$BC$$, the measure of $$\angle PP'C$$ is          ; when the length of segment $$CP'$$ is minimized, the measure of $$\angle PP'C$$ is         .

Answer: $$120 \circ$$     $$75 \circ$$


As shown in the figure, given $$\angle M O N$$, take points $$P_{1}$$, $$P_{3}$$, $$P_{5} \ldots$$ successively on side $$O N$$, and points $$P_{2}$$, $$P_{4}$$, $$P_{6} \ldots$$ successively on side $$O M$$, such that $$O P_{1} = P_{1} P_{2} = P_{2} P_{3} = P_{3} P_{4} = P_{4} P_{5} \ldots$$, forming isosceles triangles $$\triangle O P_{1} P_{2}$$, $$\triangle P_{1} P_{2} P_{3}$$, $$\triangle P_{2} P_{3} P_{4}$$, $$\triangle P_{3} P_{4} P_{5} \ldots$$

(1) If $$\angle M O N = 30^\circ$$, the last isosceles triangle obtained is            ; (2) If the above procedure is followed, and the last isosceles triangle obtained is $$\triangle P_{3} P_{4} P_{5}$$, then the range of possible values for $$\angle M O N$$, denoted as $$\alpha$$, is            .

Answer: △$$P_{1} P_{2} P_{3}$$     $$18 \circ \leq \alpha < 22.5 \circ$$


As shown in the figure, in $$R t \triangle A C B$$, $$\angle A C B = 90 \circ$$, $$\angle B = 30 \circ$$, $$B C = 4 \sqrt{3}$$, draw an arc with center at point C and radius equal to $$A C$$, intersecting $$A B$$ and $$B C$$ at points D and E respectively; draw another arc with center at point E and radius equal to $$C E$$, intersecting $$A B$$ at point F and intersecting $$\overset{⌢}{A E}$$ at point G. The area of the shaded region in the figure is          .

Answer: $$4 \pi - 4 \sqrt{3}$$


As shown in the figure, in quadrilateral ABCD, AD = 4, CD = 3, ∠ABC = ∠ACB = ∠ADC = 45°, then the length of BD is       .

Answer: $$\sqrt{41}$$


As shown in the figure, in sector $$OAB$$, $$\angle AOB = 120^\circ$$, radius $$OA = 2$$, point C is a point on $$OA$$, connect $$BC$$, fold the sector along $$BC$$ such that point A lands at point D on the extension of $$BO$$, connect $$CD$$, then the area of the shaded region in the figure is          (answer in terms of $$\pi$$).

Answer: $$\frac{4}{3} \pi - 3 + \sqrt{3}$$


As shown in the figure, in $$\triangle A B C$$, $$A C = B C = 3$$, $$\angle C = 90 \circ$$, point D lies on side $$B C$$ (not coinciding with points B or C), connect $$A D$$, point E lies on side $$A B$$, $$\angle E D B = \angle A D C$$. It is known that point H lies on ray $$A C$$, connect $$E H$$ intersecting segment $$A D$$ at point G. When $$C H = 1$$, and $$\angle A E H = \angle B E D$$, then $$\frac{B E}{A B} =$$        .

Answer: $$\frac{1}{2}$$ or $$\frac{1}{5}$$


As shown in the figure, the angle formed by lines $$l_{1}$$ and $$l_{2}$$, $$\angle B_{1} O A_{1} = 30 \circ$$, draw $$A_{1} B_{1} \bot l_{1}$$ intersecting line $$l_{2}$$ at point $$B_{1}$$, with $$O B_{1} = 2$$. Construct an equilateral triangle $$A_{1} B_{1} C_{1}$$ with $$A_{1} B_{1}$$ as a side outside $$\triangle O A_{1} B_{1}$$. Then, from point $$C_{1}$$, draw $$A_{2} B_{2} \bot l_{1}$$, intersecting lines $$l_{1}$$ and $$l_{2}$$ at points $$A_{2}$$ and $$B_{2}$$ respectively. Construct an equilateral triangle $$A_{2} B_{2} C_{2}$$ with $$A_{2} B_{2}$$ as a side outside $$\triangle O A_{2} B_{2}$$, and so on. Following this pattern, the perimeter of the 2023rd equilateral triangle $$A_{2023} B_{2023} C_{2023}$$ is           .

Answer: $$\frac{3^{2023}}{2^{2022}}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$B E$$ bisects $$\angle A B C$$, $$\angle B O D = 45 \circ$$, $$O F \bot A D$$, the following conclusions: ① $$A D$$ bisects $$\angle B A C$$; ② $$A D = O G + O F$$; ③ If $$B D = 3$$, $$A B = 12$$, then $$A G = 9$$; ④ $$S_{\triangle A C D} : S_{\triangle A B D} = A B : A C$$. Among these, the correct ones are           (fill in the serial numbers).

Answer: ①②③


As shown in the figure, in $$\triangle M N G$$, $$M N = 4 \sqrt{2}$$, $$\angle M = 75 \circ$$, $$M G = 3$$, point $$O$$ is a point inside $$\triangle M N G$$, then the minimum value of the sum of the distances from point $$O$$ to the three vertices of $$\triangle M N G$$ is        .

Answer: $$\sqrt{65}$$


As shown in the figure, given square ABCD, point M is a moving point on the extension of side BA (not coinciding with point A), and AM < AB. Triangle CBE is obtained by translating triangle DAM. If from point E, EH is drawn perpendicular to AC, with H as the foot of the perpendicular, then the following conclusions hold:
① When point M moves to a position such that ∠DHC = 60°, then 2BE = DM;
② Regardless of where point M moves, DM = √2 HM;
③ During the motion of point M, quadrilateral CEMD cannot be a rhombus;
④ Regardless of where point M moves, ∠CHM is always greater than 135°.
Which of the above conclusions are correct? (Fill in all the correct conclusion numbers.)

Answer: ①②③④


Given the isosceles right triangle $$A B C$$ and the isosceles right triangle $$A D E$$ with right-angle vertex $$A$$ coinciding, $$A B = A C = 3 \sqrt{2}$$, $$A D = A E = 2$$. Connect $$B E$$ and $$C D$$. Rotate $$\triangle A D E$$ around point $$A$$ in the plane; the rotated triangle is $$A E^{'} D^{'}$$. If point $$M$$ is the midpoint of $$B E$$, and when points $$E^{'}$$, $$D^{'}$$, and $$C$$ are collinear, the length of segment $$A M$$ is     .

Answer: $$2 + \frac{\sqrt{2}}{2}$$ or $$2 - \frac{\sqrt{2}}{2}$$


As shown in the figure, construct a square $ABDE$ on the hypotenuse $AB$ of the right triangle $ABC$, on the same side as $\triangle ABC$. Let $O$ be the center of the square, and connect $OC$. If $AB = 13$ and $AC = 5$, then the length of $OC$ is          .

Answer: $$\frac{7 \sqrt{2}}{2}$$


As shown in the figure, in sector $$OAB$$, $$\angle AOB = 90^\circ$$, radius $$OA = 8$$. Point $$F$$ is located at $$\frac{1}{3}$$ of arc $$\overset{⌢}{AB}$$, closer to point $$A$$. Points $$C$$ and $$D$$ lie on segments $$OA$$ and $$OB$$ respectively, with $$CD = 8$$. $$E$$ is the midpoint of $$CD$$. Connect $$EF$$ and $$BE$$. As $$CD$$ slides (with its length remaining constant), when $$EF$$ takes its minimum value, the perimeter of the shaded region is           .

Answer: $$4 + 4 \sqrt{3} + \frac{8}{3} \pi$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$C B = 7$$, $$A C = 9$$, a circle $$\bigodot C$$ with center $$C$$ and radius 3 is drawn. Point $$P$$ is a moving point on $$\bigodot C$$. Connect $$A P$$ and $$B P$$. The minimum value of $$\frac{1}{3} A P + B P$$ is      .

Answer: $$5 \sqrt{2}$$


A triangular piece of material is shown in the figure, $$\angle B = 30^\circ, \angle C = 90^\circ$$, $$AC = 4$$, cut out a rectangle $$CDEF$$ from this material, where points D, E, F lie on $$BC, AB, AC$$ respectively, the maximum area of the rectangle $$CDEF$$ that can be cut out is      .

Answer: $$4 \sqrt{3}$$


As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C$$, $$B D$$ intersect at point O. Points E and F are two moving points on $$O D$$ and $$O C$$ respectively, with $$E F = 4$$. P is the midpoint of $$E F$$. Connect $$O P$$, $$P C$$, $$P D$$. If $$A C = 12$$, $$B D = 16$$, then the minimum value of $$P C + \frac{1}{4} P D$$ is                .

Answer: $$\frac{\sqrt{145}}{2}$$/$$\frac{1}{2} \sqrt{145}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle B A C = 30 \circ$$, point $$D$$ is the midpoint of $$A C$$, connect $$B D$$, fold $$\triangle A B D$$ along $$B D$$ to obtain $$\triangle A^{'} B D$$, connect $$A A^{'}$$, $$C A^{'}$$, if $$B C = 1$$, then the length of $$A^{'} C$$ is      .

Answer: $$\frac{3 \sqrt{7}}{7}$$


As shown in the figure, in the right triangle $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, $$A B = B C = 1$$, points $$D$$ and $$E$$ are moving points on sides $$A B$$ and $$B C$$ respectively, satisfying $$A D = B E$$. Connect $$A E$$ and $$C D$$, then the minimum value of $$A E + C D$$ is         .

Answer: $$\sqrt{5}$$


As shown in the figure, in square $$ABCD$$, $$\triangle BPC$$ is an equilateral triangle, and the extensions of $$BP$$ and $$CP$$ intersect $$AD$$ at points $$E$$ and $$F$$, respectively. Connect $$BD$$ and $$DP$$; $$BD$$ intersects $$CF$$ at point $$H$$. The following conclusions are given: ① $$AE = \frac{1}{2} FC$$; ② $$\angle PDE = 15^\circ$$; ③ $$\frac{S_{\triangle DHC}}{S_{\triangle BHC}} = \frac{\sqrt{3}}{3}$$; ④ $$DE^2 = PF \cdot FC$$. Among these, the correct ones are      (fill in the serial numbers)

Answer: ①②③④


As shown in the figure, in the equilateral triangle $$\triangle ABC$$ with side length $$4 \sqrt{3}$$, moving points D and E lie on sides $$BC$$ and $$AC$$ respectively, such that $$AE = CD$$. Connect $$BE$$ and $$AD$$, intersecting at point P. Then the minimum value of $$CP$$ is          .

Answer: 4


As shown in the figure, in rectangle $$ABCD$$, $$E$$ is the midpoint of $$BC$$. After folding $$\triangle ABE$$ to obtain $$\triangle AFE$$, point $$F$$ lies inside the rectangle. Extend $$AF$$ to intersect $$CD$$ at point $$H$$. If $$AD = 4$$ and $$CH = \frac{4}{3}$$, then the length of the fold line $$AE$$ is      .

Answer: $$\sqrt{13}$$


As shown in the figure, the side length of square ABCD is 4, and point E is any point on diagonal BD (not coinciding with B or D). Connect AE, and from point E draw EF ⊥ AE, intersecting segment BC at point F. Construct rectangle AEFG with AE and EF as adjacent sides. Connect BG. The following four conclusions are given:
① AE = EF; ② CD - BF = (√2)/2 * BE;
③ Let the perimeter of quadrilateral AGBE be m, then 8√2 ≤ m < 8 + 4√2;
④ When BF = 1, the area of triangle AGB is 3.
Among these, the correct conclusions are            . (Fill in all the serial numbers of the correct conclusions)

Answer: ①③④


As shown in the figure, the side length of square $$A B C D$$ is 4, O is the intersection point of the diagonals, M and N are moving points on sides $$A D$$ and $$C D$$ respectively, and $$A M = C N$$. Connect $$O M$$ and $$B N$$; then the minimum value of $$O M + B N$$ is                  .

Answer: $$2 \sqrt{10}$$


As shown in the figure, in the rhombus $$A B C D$$, $$\angle B = 60^\circ$$, point E and point F are moving points on segments $$A B$$ and $$A D$$ respectively, and $$B E = A F$$. Let the side length of rhombus $$A B C D$$ be $$2 a$$, and the perimeter of $$\triangle A E F$$ be $$m$$. Then the range of the perimeter $$m$$ of $$\triangle A E F$$ is      . (Expressed in terms of $$a$$)

Answer: $$2 a + \sqrt{3} a \leq m < 4 a$$


Given that quadrilateral $$A B C D$$ is a parallelogram, $$A B = 15$$, $$A D = 20$$, $$A H \bot B C$$ at point H, $$A H = 12$$, point E is a point on segment $$C D$$, connect $$A E$$, fold $$\triangle A D E$$ along $$A E$$ to obtain $$\triangle A F E$$, point D lands at point F on the extension of $$B C$$, $$A F$$ intersects $$C D$$ at point G, then $$S_{\triangle A G E} =$$      .

Answer: $$\frac{128}{3}$$


As shown in the figure, points E and F are on sides BC and CD of rectangle ABCD, respectively. Triangle CEF is folded along line EF to obtain triangle GEF, and then triangle BEG is folded along line BG; the image of point E, point H, falls exactly on diagonal BD. If at this time points F, G, and H are collinear, and segment HF is also symmetric to segment HD with respect to some line, then the value of $$\frac{BD}{EF}$$ is      .

Answer: $$2 + \sqrt{3}$$


As shown in the figure, in the isosceles right triangle $$ABC$$, $$\angle ABC = 90^\circ$$, $$AB = 6$$, segment $$PQ$$ moves along the hypotenuse $$AC$$, and $$PQ = 2$$. Connect $$BP$$ and $$BQ$$. What is the minimum perimeter of $$\triangle BPQ$$?

Answer: $$2 \sqrt{19} + 2$$/$$2 + 2 \sqrt{19}$$


As shown in the figure, in $$\triangle$$ABC, AD and CE are medians. If the area of quadrilateral BDFE is 6, then the area of $$\triangle$$ABC is         .

Answer: 18


As shown in the figure, in isosceles triangle ABC, AB = AC = 5, BC = 8. E is a moving point on BC. Triangle ABE is folded along AE to triangle ADE. Then side AC is folded to coincide with AD, with crease AF. When triangle DEF is isosceles, the length of BE is ______.

Answer: $$\frac{5}{2}$$ or $$\frac{25}{8}$$ or $$\frac{7}{4}$$.


As shown in the figure, in quadrilateral ABCD, $$\angle ABC = 135^\circ, \angle BCD = 120^\circ, CD = 2\sqrt{3}, AB = \sqrt{2}, BC = 3 - \sqrt{3}$$, then the perimeter of quadrilateral $$ABCD$$ is           .

Answer: $$3 + \sqrt{2} + \sqrt{3} + 2 \sqrt{5}$$


As shown in the figure, in square $$ABCD$$, $$AB = 2$$, diagonals $$AC$$ and $$BD$$ intersect at point O. Points E and F are moving points on sides $$BC$$ and $$CD$$ respectively (points E and F do not coincide with endpoints of segments $$BC$$ and $$CD$$) and satisfy $$BE = CF$$. Connect $$OE$$, $$OF$$, and $$EF$$. As points E and F move, the following four conclusions are given: ① $$\triangle OEF$$ is an isosceles right triangle; ② The minimum area of $$\triangle OEF$$ is $$\frac{1}{2}$$; ③ There exists at least one $$\triangle ECF$$ such that the perimeter of $$\triangle ECF$$ is $$2 + \sqrt{3}$$; ④ The area of quadrilateral $$OECF$$ is 1. Write the serial numbers of the correct conclusions        .

Answer: ①②③④


As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 8$$, points $$E, F$$ are two moving points on sides $$A B$$ and $$B C$$ respectively, and $$E F = 2$$, point $$G$$ is the midpoint of $$E F$$, point $$H$$ is a moving point on side $$A D$$, connect $$C H$$ and $$G H$$, then the minimum value of $$G H + C H$$ is           .

Answer: $$2 \sqrt{41} - 1$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A = 30^\circ$$, $$\angle A C B = 90^\circ$$, $$B C = 2$$, D is a moving point on $$A B$$, rotate segment $$C D$$ counterclockwise by 90° about point C to obtain segment $$C E$$, connect $$B E$$, then the minimum value of $$B E$$ is     .

Answer: $$\sqrt{3} - 1$$/$$- 1 + \sqrt{3}$$


In parallelogram $$A B C D$$, $$\angle A B C$$ is an acute angle. Fold $$C D$$ along line $$l$$ onto the line containing $$A B$$, with corresponding points $$C^{'}$$ and $$D^{'}$$. If $$A C^{'} : A B : B C = 1 : 3 : 7$$, and from point F draw a perpendicular to $$A B$$ intersecting it at E, then $$\frac{B E}{B F} =$$         .

Answer: $$\frac{2}{7}$$ or $$\frac{4}{7}$$/$$\frac{4}{7}$$ or $$\frac{2}{7}$$


As shown in the figure, in △ABC, AC = 2 + 2$$\sqrt{3}$$, ∠BAC = 45°, ∠ACB = 30°. Rotate △ABC counterclockwise about point B to obtain $$\triangle A_{1} B C_{1}$$, where point E is the midpoint of segment AB, and point P is a moving point on segment AC. During the counterclockwise rotation of △ABC about point B, the corresponding point of P is point $$P_{1}$$. The maximum value of segment $$E P_{1}$$ is     , and the minimum value is        .

Answer: $$4 + \sqrt{2}$$/$$\sqrt{2} + 4$$     $$2 - \sqrt{2}$$/$$- \sqrt{2} + 2$$


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle C = 90 \circ$$, $$A B = B C = 10 \sqrt{3}$$, $$A D = C D = 10$$, points $$E$$ and $$F$$ lie on sides $$A D$$ and $$B C$$ respectively, connect $$E F$$, point $$G$$ is the midpoint of $$E F$$, connect $$A G$$, if $$A E = 6$$, then the minimum value of $$A G$$ is      .

Answer: 9


As shown in the figure, in square $$ABCD$$ with side length 2, points $$E$$ and $$F$$ lie on sides $$AB$$ and $$BC$$ respectively, $$AF \bot DE$$, with foot of perpendicular at point $$G$$. Construct rectangle $$DG FH$$ with sides $$DG$$ and $$GF$$. If the area of the shaded region in the figure is 3, then the area of rectangle $$DG FH$$ is     .

Answer: 3


As shown in the figure, in quadrilateral $$A B C D$$, $$A B = B C = \frac{5 \sqrt{2}}{2}$$, $$C D = 3$$, $$\angle A B C = \angle A D C = 90 \circ$$, $$E$$ is a point on $$B D$$ such that $$\angle A E C = 135 \circ$$, then $$A E =$$      .

Answer: $$\sqrt{10}$$


As shown in the figure, in the rhombus $$A B C D$$, $$A B = 2 \sqrt{3}$$, $$\angle A B C = 60 \circ$$, point E is a moving point on diagonal $$B D$$ (not coinciding with point B), and $$B E < \frac{1}{2} B D$$, connect $$C E$$ intersecting the extension of $$D A$$ at point F.

①$$\angle A F E = \angle B A E$$;② When $$\triangle A E F$$ is a right triangle, $$B E = 2$$;③ When $$\triangle A E F$$ is an isosceles triangle, $$\angle A F C = 20 \circ$$ or $$\angle A F C = 40 \circ$$;④ Connect $$B F$$, when $$B E = C E$$, $$F C$$ bisects $$\angle A F B$$.Which of the above conclusions are correct? (Fill in the correct serial numbers.)

Answer: ①②③④


As shown in the figure, in the rhombus $$A B C D$$, $$\angle B = 60 \circ$$, $$A B = 4$$, $$E$$ and $$F$$ are points on sides $$A B$$ and $$B C$$ respectively. Fold $$\triangle E B F$$ along $$E F$$ so that the image of point $$B$$, denoted $$B'$$, lies on side $$A D$$. If $$A E = A B'$$, then the length of $$C F$$ is      .

Answer: $$4 - 2 \sqrt{3}$$/$$- 2 \sqrt{3} + 4$$


As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram. Extend $$A D$$ to point E such that $$D E = A D$$, and $$B E \bot D C$$. If $$\triangle A D B$$ is an equilateral triangle with side length 3, and points P, M, N move along segments BE, BC, and CE respectively, then the minimum value of $$P M + P N$$ is      .

Answer: $$\frac{3 \sqrt{3}}{2}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$A D \bot C D, B C \bot C D$$, point $$E$$ is the midpoint of $$A B$$, $$B C = C D = 6, A B = 2 \sqrt{10}$$, point $$F$$ lies on side $$C D$$, and the area of quadrilateral $$A E F D$$ is 9. Find the value of $$\frac{D F}{F C}$$.

Answer: $$\frac{1}{4}$$/$$0 . 25$$


As shown in the figure, in $$\triangle$$ABC, $$\angle B A C = 60 \circ$$, $$\angle A B C = 45 \circ$$, AD bisects $$\angle B A C$$ and intersects BC at point D, P is a moving point on line AB. Connect DP, and construct parallelogram DPQB with DP and DB as adjacent sides. Connect CQ. If $$A C = 6$$, then the minimum value of CQ is      .

Answer: $$3 + 3 \sqrt{3}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$A C = B C, \angle A C B = 90 \circ$$, if $$\angle C A E = 2 \angle A B D, A E = 5, E C = 3$$, then the length of $$C D$$ is       .

Answer: 2


As shown in the figure, given equilateral triangle $$\triangle ABC$$ with side length 6, point D lies on the extension of $$BC$$ such that $$CD = 4$$, and point E is a moving point on line $$AC$$. Connect $$DE$$. Let F and G be the midpoints of $$AB$$ and $$DE$$, respectively. Connect $$FG$$. What is the minimum length of segment $$FG$$?

Answer: $$\frac{5}{2} \sqrt{3}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$A D$$ is an angle bisector of $$\triangle A B C$$, $$E$$ is the midpoint of $$A D$$, and $$B E$$ is connected. If $$B E = B C$$ and $$C D = 2$$, then $$B D =$$      .

Answer: $$\frac{\sqrt{17} + 1}{2}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle B C D = 90 \circ$$, diagonals $$A C, B D$$ intersect at point $$O$$. If $$A B = A C = 5, B C = 6, \angle A D B = 2 \angle C B D$$, then the length of $$A D$$ is          .

Answer: $$\frac{\sqrt{97}}{3}$$/$$\frac{1}{3} \sqrt{97}$$


As shown in the figure, $$\triangle A B C$$ is inscribed in $$\bigodot O$$, diameter $$A C$$ intersects chord $$B D$$ at point E, and the extension of $$B D$$ intersects the tangent at point C at point F. Connect $$C D$$. If $$B D = \frac{8}{3} D E$$, $$C F = 3$$, $$D F = 1$$, then $$B F =$$     , $$A B =$$        .

Answer: 9     $$\frac{5}{14} \sqrt{154}$$/$$\frac{5 \sqrt{154}}{14}$$


As shown in the figure, a large equilateral triangle $$ABC$$ is formed by three congruent triangles ($$\triangle ABE$$, $$\triangle BCF$$, $$\triangle CAD$$) and a small equilateral triangle $$DEF$$ in the center. Connect $$BD$$ and extend it to intersect $$AC$$ at point $$G$$. If $$AE = ED = 2$$, then:
(1) The measure of $$\angle FDB$$ is      ;
(2) The length of $$DG$$ is      .

Answer: $$30 \circ$$     $$\frac{4 \sqrt{3}}{5}$$


As shown in the figure, in $$\triangle A B C$$, point $$D$$ is a point on side $$A C$$, with $$C D : A D = 1 : 2$$. Connect $$B D$$, point $$E$$ is a point on segment $$B D$$, with $$B E : E D = 1 : 3$$. Connect $$A E$$, point $$F$$ is the midpoint of segment $$A E$$. Connect $$C F$$, intersecting segment $$B D$$ at point $$G$$. If the area of $$\triangle A B C$$ is 12, then the area of $$\triangle E F G$$ is        .

Answer: $$\frac{9}{4}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, construct $$\triangle A C D$$ with side $$A C$$ such that $$A D = A C$$, point E is a point on $$B C$$, connect AE, $$\angle B A E = \frac{1}{2} \angle C A D$$, connect $$D E$$ . Which of the following conclusions is correct? (Fill in the serial number)
① $$A C \bot D E$$; ② $$\angle A D E = \angle A C B$$; ③ If $$C D / / A B$$, then $$A E \bot A D$$; ④ $$D E = C E + 2 B E$$.

Answer: ②③④


As shown in the figure, the circle $$\bigodot O$$ with diameter $$AB$$ is perpendicular to chord $$CD$$ at point $$E$$. From point $$D$$, draw $$DF \bot AC$$ intersecting at point $$F$$, intersecting $$\bigodot O$$ at point $$G$$, and intersecting $$AB$$ at point $$H$$. Connect $$AG$$, $$CG$$, and $$EF$$. Given $$CG = \sqrt{5}$$ and $$AH = 3$$, then $$AG =$$     , and the length of segment $$EF =$$        .

Answer: 3     2


As shown in the figure, in $$R t \triangle A B C$$, $$\angle B A C = 90 \circ, A B = 3, B C = 5$$, point P is any point on side $$B C$$, connect $$P A$$, construct parallelogram $$P A Q C$$ with $$P A$$ and $$P C$$ as adjacent sides, connect $$P Q$$, then the minimum length of $$P Q$$ is         .

Answer: $$\frac{12}{5}$$/2.4


As shown in the figure, in △ABC, AB = 5, AC = 8, BO and CO bisect ∠ABC and ∠ACB respectively. Draw DE ∥ BC through point O. Then the perimeter of △ADE is                .

Answer: 13


As shown in the figure, points A and B are two points on $$\bigodot O$$, connect $$AB$$, diameter $$CD$$ is perpendicular to $$AB$$ at point E, point F is on $$\bigodot O$$, connect $$AF$$, $$BF$$, draw a perpendicular from point A to $$BF$$, intersecting $$BF$$ at point G and intersecting $$\bigodot O$$ at point H. If $$AE = 3$$, $$CD = 4\sqrt{3}$$, $$GH = \sqrt{2}$$, then the length of $$OE$$ is       , and the length of $$AF$$ is          .

Answer: $$\sqrt{3}$$     $$2 \sqrt{10}$$


As shown in the figure, in rectangle ABCD, AB = 4, BC = 3, E is a moving point on side AB, and a square DEFG is constructed to the right with DE as a side. Connect CF; then the minimum value of CF is      .

Answer: $$5 \sqrt{2}$$


As shown in the figure, the square $$A B C D$$ has side length 4, $$L$$ is the midpoint of $$C D$$, and $$Y, B, D$$ lie on $$\bigodot C$$, the maximum value of $$\left|\sqrt{2} L Y - Y A\right|$$ is, and the minimum value of $$2 \sqrt{2} L Y + Y A$$ is

Answer: $$2 \sqrt{2}$$     $$4 \sqrt{5}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$A B = B C, \angle A B C = \angle C D A = 90^{\circ}, B E \bot A D$$ at $$E, S_{A B C D} = 10$$, then the length of $$B E$$ is

Answer: $$\sqrt{10}$$


As shown in the figure, $$\triangle A B C, \triangle C D E$$ are both equilateral triangles. Rotate $$\triangle C D E$$ about point $$C$$ such that points $$A, D, E$$ lie on the same straight line. Connect $$B E$$. If $$B E = 1, A E = 4$$, then the length of $$C E$$ is      .

Answer: 3


As shown in the figure, in rectangle $$A B C D$$, point $$E$$ is the midpoint of $$A D$$, and point $$F$$ is a point on $$A B$$. After folding $$\triangle A E F$$ along $$E F$$, point $$A$$ lands exactly at point $$G$$ on $$C F$$. Draw $$F H \parallel A D$$ intersecting $$E G$$ at point $$H$$. If $$A B = 16$$ and $$A D = 24$$, then $$G H =$$      .

Answer: $$\frac{21}{8}$$/$$2 \frac{5}{8}$$/2.625


As shown in the figure, point C is a moving point outside the straight line AB, with AB = 6. Connect CA and CB. Points D and E are the midpoints of AB and BC, respectively. Connect AE and CD, intersecting at point F. When the area of quadrilateral BEFD is 5, the minimum length of segment AC is      .

Answer: 5


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, $$AD$$ and $$BC$$ are tangent to $$\bigodot O$$ at points $$A$$ and $$B$$ respectively, $$CD$$ passes through a point $$E$$ on $$\bigodot O$$, $$AD = DE$$, if $$AB = 12$$, $$BC = 4$$, then the length of $$AD$$ is      .

Answer: $$9$$


As shown in the figure, in rectangle $$ABCD$$, $$AB = 6$$, $$AD = 4$$, point $$O$$ is a point on side $$AB$$ such that $$AO = 2$$, and point $$E$$ is a moving point on side $$BC$$. Rotate segment $$OE$$ clockwise around point $$O$$ by $$120^{\circ}$$ to obtain segment $$OE'$$, which intersects side $$AD$$ at point $$F$$. Connect $$EF$$.

(1) When point $$E$$ coincides with point $$B$$, the area of $$\triangle EOF$$ is          . (2) When point $$E$$ moves along side $$BC$$, the minimum area of $$\triangle EOF$$ is          .

Answer: $$4 \sqrt{3}$$     $$\frac{8 \sqrt{3}}{3}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$A B = 6$$, $$\angle A C B = 30 \circ$$, $$E$$ is the midpoint of $$B C$$, and $$\triangle A B C$$ is folded along side $$A C$$ to obtain $$\triangle A F C$$. Points $$M$$ and $$N$$ are two moving points on side $$A C$$ such that $$M N = 2$$. The minimum perimeter of quadrilateral $$B E N M$$ is        .

Answer: $$2 + 3 \sqrt{3} + \sqrt{67}$$


As shown in the figure, in the given $$\bigodot O$$, the perpendicular distance from the center to chord $$AB$$ is $$OH = 6$$, $$CD = 16$$, and point $$E$$ lies on chord $$CD$$ such that $$OE = ED = 5$$. The maximum area of $$\triangle EAB$$ is  , and at this time, the length of $$DH$$ is     .

Answer: $$22 \sqrt{11}$$     $$2 \sqrt{53}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C > 90 \circ$$, $$A B = \sqrt{13}$$, $$B C = 10$$, point D is the midpoint of $$B C$$, point E lies on $$A C$$, and $$\triangle C D E$$ is folded along $$D E$$ such that point C lands exactly at point F on the extension of $$B A$$. Connect $$A D$$, given that $$S_{\Delta E D F} = \frac{5}{2}$$, find the length of $$A D$$.

Answer: $$2 \sqrt{2}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A = m \circ$$, the angle bisectors of $$\angle A B C$$ and $$\angle A C D$$ intersect at point $$A_{1}$$, yielding $$\angle A_{1}$$; the angle bisectors of $$\angle A_{1} B C$$ and $$\angle A_{1} C D$$ intersect at point $$A_{2}$$, yielding $$\angle A_{2}$$; .. the angle bisectors of $$\angle A_{2020} B C$$ and $$\angle A_{2020} C D$$ intersect at point $$A_{2021}$$, yielding $$\angle A_{2021}$$; the angle bisectors of $$\angle A_{2021} B C$$ and $$\angle A_{2021} C D$$ intersect at point $$A_{2022}$$, yielding $$\angle A_{2022}$$. Then $$\angle A_{2022} =$$        degrees.

Answer: $$\frac{m}{2^{2022}}$$


As shown in the figure, in the right triangle $$\text{Rt} \triangle A B C$$, $$\angle C = 90^\circ$$, $$AC = 3$$, $$BC = 4$$, there is a point P on segment $$BC$$ (not coinciding with B or C). Connect $$AP$$, and fold $$\triangle ACP$$ along $$AP$$ to obtain $$\triangle ADP$$. Connect $$CD$$ and $$BD$$. When $$\triangle CDB$$ is an isosceles triangle, the length of $$CP$$ is        .

Answer: $$\frac{6}{5} \sqrt{5}$$ or $$\frac{9 - 3 \sqrt{5}}{2}$$


As shown in the figure, in the square $$ABCD$$ with side length 2, points E and F are moving points on sides $$AD$$ and $$CD$$ respectively (not coinciding with the endpoints). Connect $$BE$$ and $$BF$$, intersecting the diagonal $$AC$$ at points P and Q respectively. As points E and F move, $$\angle EBF = 45^\circ$$ always holds. Connect $$EF$$, $$PF$$, and $$PD$$. The following conclusions: ① $$PB = PD$$; ② $$\angle EFD = 2\angle FBC$$; ③ $$PQ = PA + CQ$$; ④ $$\triangle BPF$$ is an isosceles right triangle; ⑤ If a perpendicular $$BH \bot EF$$ is drawn from point B with foot at H, and $$DH$$ is connected, then the minimum value of $$DH$$ is $$2\sqrt{2} - 2$$. The sequence numbers of all correct conclusions are    .

Answer: ①②④⑤


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, chord $$CD \bot AB$$ at point $$E$$, point $$F$$ lies on the circle, and $$\overset{⌢}{DF} = \overset{⌢}{CD}$$, $$BE = 2$$, $$CD = 8$$, $$CF$$ intersects $$AB$$ at point $$G$$. Then the length of chord $$CF$$ is                  , and the length of $$AF$$ is                     .

Answer: $$\frac{48}{5}$$     $$\frac{4 \sqrt{5}}{5}$$/$$\frac{4}{5} \sqrt{5}$$


Given $$\triangle A B C$$, $$A B = A C$$, $$A D \bot B C$$, point F lies on $$A C$$, construct $$E F \bot A B$$ at E, intersecting the extension of $$B C$$ at G, connect $$E D$$, $$\angle G F C = 2 \angle E D A$$, $$D H = C G = 2$$, then the length of $$A F$$ is      .

Answer: $$\frac{4}{3} \sqrt{5}$$/$$\frac{4 \sqrt{5}}{3}$$


As shown in the figure, it is given that $$AB = 5$$, the radius of $$\bigodot B$$ is 2, point C lies on $$\bigodot B$$, and segment $$AC$$ is connected. Rotate segment $$AC$$ clockwise around point A by $$60^\circ$$ to obtain segment $$AD$$. When point C moves along $$\bigodot B$$, the maximum distance from point D to line $$AB$$ is        .

Answer: $$\frac{5}{2} \sqrt{3} + 2$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A B = A C$$, rotate $$A C$$ clockwise around point C by $$60 \circ$$ to obtain $$C D$$, connect BD intersecting $$A C$$ at E, then $$\frac{A E}{E D} =$$      .

Answer: $$\frac{3 \sqrt{2} - \sqrt{6}}{6}$$


In rectangle $$A B C D$$, $$A B = \sqrt{3}$$, $$B C = 1$$, fold $$\triangle A B C$$ along $$A C$$ to obtain $$\triangle A E C$$, F is a point on $$D C$$, connect $$E F$$, if $$s i n \angle A E F = \frac{\sqrt{21}}{7}$$, then the length of segment $$E F$$ is        .

Answer: $$\frac{\sqrt{21}}{9}$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, $$C$$ is a point on the circle, and $$\angle AOC = 120^\circ$$. The radius of $$\bigodot O$$ is $$4$$. Point $$P$$ is a moving point on the circle, and $$Q$$ is the midpoint of $$AP$$. What is the maximum length of $$CQ$$?

Answer: $$2 + 2 \sqrt{7}$$/$$2 \sqrt{7} + 2$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, extend $$B C$$ to point E such that $$B E = 8$$. A moving point P starts from point B and moves toward point A along the path $$B C - C D - D A$$ at a speed of 2 units per second. Let the motion time of point P be $$t$$ seconds. When $$\triangle A B P$$ and $$\triangle D C E$$ are congruent, the value of $$t$$ is          .

Answer: 1 or 7


As shown in the figure, in $$\triangle A B C$$, points D and E are points on sides $$A B$$ and $$A C$$ respectively. Fold $$A D$$ and $$B D$$ along $$D E$$ and $$D C$$ to $$A^{'} D$$. Given that $$\angle A^{'} C A = 36 \circ$$ and $$\angle B + \frac{\angle A^{'} E C}{2} = 90 \circ$$, then $$\angle A^{'} D C$$ is           $$\circ$$.

Answer: 54


As shown in the figure, in $$\triangle A B C$$, point $$D$$ is a point on side $$A B$$, with $$A D : D B = 3 : 1$$, connect $$C D$$, point $$E$$ is a point on segment $$A C$$, with $$A E : E C = 1 : 2$$, connect $$B E$$, $$C D$$ and $$B E$$ intersect at point $$F$$, if $$A C = 8$$, $$B C = 9$$, then the maximum value of the sum of the areas of $$\triangle B D F$$ and $$\triangle C E F$$ is      .

Answer: $$17$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$\angle B = 30 \circ$$, $$B C = 4$$, points P and D are moving points on $$B C$$ and $$A B$$ respectively, then the minimum value of $$A P + D P$$ is          .

Answer: 4


As shown in the figure, $$\triangle A B C$$ and $$\triangle C E F$$ are both isosceles right triangles, $$\angle B A C = \angle C E F = 90 \circ$$, and point E lies on side $$A C$$. Rotate $$\triangle C E F$$ counterclockwise about point C by $$\alpha \left(\right. 0 \circ < \alpha < 180 \circ \left.\right)$$; during the rotation, line $$E F$$ intersects line $$A C$$ and line $$B C$$ at points M and N, respectively. If $$\triangle C M N$$ is an isosceles triangle, then the value of $$\alpha$$ is      .

Answer: $$22.5 \circ$$ or $$45 \circ$$ or $$112.5 \circ$$


As shown in the figure, square $$ABCD$$ is folded along line $$EF$$ such that the image of point $$B$$, denoted $$M$$, lies on side $$AD$$, and point $$C$$ lands at point $$N$$. Segment $$MN$$ intersects $$CD$$ at point $$P$$. The crease intersects sides $$AB$$ and $$CD$$ at points $$E$$ and $$F$$, respectively. Connect $$BM$$. If $$\frac{DP}{CP} = \frac{1}{2}$$, then the value of $$\frac{AE}{BE}$$ is        .

Answer: $$\frac{12}{13}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle B = 30 \circ$$, moving points $$M$$ and $$N$$ are on $$B C$$ and $$A B$$ respectively, and $$A N = 2 B M > 0$$. Connect $$A M$$ and $$C N$$. If $$A C = 1$$, then the minimum value of $$C N + 2 A M$$ is         .

Answer: $$\sqrt{17}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 5, B C = 9$$, E is a point on side $$A B$$, $$A E = 2$$, F is a moving point on line $$B C$$, rotate line segment $$E F$$ counterclockwise by $$90 \circ$$ about point E to obtain segment $$E G$$, connect $$C G, D G$$, then the minimum value of $$C G + D G$$ is        .

Answer: 13


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, $$A B = 2$$, $$A H \bot B C$$ at point $$H$$, and an equilateral triangle $$\triangle A H M$$ is constructed to the left using $$A H$$ as a side. Let $$Q$$ be a moving point on segment $$A M$$. Connect $$B Q$$ and $$Q H$$. Then the minimum perimeter of $$\triangle Q B H$$ is                .

Answer: $$\sqrt{7} + 1$$


As shown in the figure, in rectangle ABCD, AD = 6, AB = 8. Point E is a moving point on side DC. △AD'E is symmetric to △ADE with respect to line AE. When △CD'E is a right triangle, the length of DE is  .

Answer: 3 or 6


As shown in the figure, in the isosceles right triangle $$\triangle ABC$$, $$\angle ACB = 90^\circ$$, point D lies on side $$AC$$ (excluding points A and C), connect $$BD$$, and construct an adjacent isosceles right triangle $$\triangle BDE$$ with $$DB$$ as one leg to the right side, $$\angle BDE = 90^\circ$$, and connect $$CE$$. If $$AD = 2$$, $$CE = \sqrt{5}$$, then the length of segment $$BE$$ is    .

Answer: $$2 \sqrt{5}$$


As shown in the figure, in the isosceles right triangle $$ABC$$, $$AC = BC = 1$$, point P lies on the semicircle with hypotenuse $$AB$$ as its diameter, M is the midpoint of $$PC$$, when point P moves along the semicircle from point A to point B, the length of the path traced by point M is       .

Answer: $$\frac{\sqrt{2}}{4} \pi$$


As shown in the figure, G is a point on the diagonal AC of square ABCD. Connect BG, and draw EF perpendicular to BG, intersecting AB at point E and CD at point F. If AG = 2√2 and CF = 2, then the area of triangle EGB is     .

Answer: 5


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$A D = 10$$, point E lies on $$A D$$ such that $$D E = 2$$. Point G is the midpoint of $$A E$$, and point P is a moving point on side $$B C$$. F is the midpoint of $$E P$$. Find the minimum value of $$G F + E F$$       .

Answer: 5


As shown in the figure, the width of rectangle $$A B C D$$ is $$8$$, and its length is $$12$$. Point $$Q$$ lies on segment $$C D$$, with $$C Q = 5$$. Point $$P$$ lies on segment $$B C$$. When $$\triangle P Q C$$ is folded along $$P Q$$, point $$C$$ lands exactly at point $$R$$ on side $$A D$$. Point $$O$$ lies on segment $$A B$$. When $$\triangle A O R$$ is folded along $$O R$$, point $$A$$ lands exactly at point $$H$$ on segment $$P R$$. Find the distance from point $$H$$ to segment $$D C$$.

Answer: $$\frac{44}{5}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points $$M$$, $$N$$ lie on sides $$C D$$, $$B C$$ respectively, and $$B N = 2 D M$$. Connect $$A M$$, draw $$N P \bot A M$$ through point $$N$$, with foot at $$P$$, and connect $$D P$$. What is the minimum length of $$D P$$?

Answer: $$2$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$A E$$ is the altitude from side $$B C$$, and points $$F$$ and $$G$$ are moving points on the altitude $$A E$$ and side $$C D$$, respectively, such that $$A F = D G$$. If $$A B = 5$$, $$B C = 4$$, $$\angle A D C = 60^\circ$$, then the minimum value of $$B F + A G$$ is      .

Answer: $$\sqrt{41}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B =\text{60} \circ$$, $$B C = 12$$. Point $$M$$ lies on side $$B C$$, and $$M C = \frac{1}{4} B C$$, ray $$C D \bot B C$$ at point $$C$$, point $$P$$ is a moving point on ray $$C D$$, and point $$N$$ is a moving point on segment $$A B$$.

(1) Does the length of segment $$M P + N P$$ have a minimum value?          . (Fill in "yes" or "no") (2) If the length of segment $$M P + N P$$ has a minimum value, directly write the length of $$B N$$; if not, provide the reason          .

Answer: yes     $$\frac{15}{2}$$


As shown in the figure, in parallelogram $$A B C D$$, $$A B = 4$$, $$A D = 6$$, $$\angle A = 120 \circ$$, points $$F$$ and $$N$$ are the midpoints of $$C D$$ and $$A B$$, respectively. Point $$E$$ moves along side $$A D$$. Fold $$\triangle E D F$$ along $$E F$$ such that point $$D$$ lands at $$D^{'}$$. Connect $$B D^{'}$$, and let point $$M$$ be the midpoint of $$B D^{'}$$. Then the minimum value of $$M N$$ is         .

Answer: $$\sqrt{7} - 1$$/$$- 1 + \sqrt{7}$$


As shown in the figure, in the isosceles triangle $$\triangle ABC$$, $$AB = AC = 20$$, $$BC = 32$$, $$\triangle ABD$$ is an equilateral triangle, and P is a moving point on the angle bisector of $$\angle BAC$$. Connect $$PC$$ and $$PD$$. The minimum value of $$PC + PD$$ is       .

Answer: 20


As shown in the figure, in rectangle paper $$ABCD$$, $$AB = 2$$, $$BC = 2\sqrt{2}$$, point $$E$$ is the midpoint of $$AB$$, and point $$F$$ is a moving point on side $$AD$$. Fold $$\triangle AEF$$ along the line $$EF$$ to obtain $$\triangle A'EF$$. Connect $$A'C$$ and $$A'D$$. When $$\triangle A'DC$$ is an isosceles triangle, the length of $$AF$$ is        .

Answer: $$\frac{\sqrt{2}}{2}$$ or 1 or $$\sqrt{2}$$


In $$\triangle A B C$$, $$A B = A C$$, $$\angle B A C = 120^\circ$$, D is a point inside $$\triangle A B C$$, and an isosceles triangle $$\triangle D A E$$ is constructed with $$A D$$ as a leg such that $$\angle D A E = \angle B A C$$. Connect $$B E$$ and $$C D$$. If $$M$$ and $$N$$ are the midpoints of $$D E$$ and $$B C$$ respectively, and $$M N = 1$$, then the length of $$C D$$ is       .

Answer: 2


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 3 \angle B$$, $$C D$$ bisects $$\angle A C B$$, $$A E \bot C D$$, with foot at point E. If $$B D = 11$$, $$A D = 5$$, then (1) $$\frac{B C}{A C} =$$      ; (2) the perimeter of $$\triangle A B C$$ is      .

Answer: $$\frac{11}{5}$$     $$16 + \frac{32}{3} \sqrt{5}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle B A C = 30 \circ$$, $$A C = 3$$, $$A D = \frac{1}{4} A B$$, points $$E$$, $$F$$ lie on segment $$A C$$, an equilateral triangle $$\triangle A E G$$ is constructed outward from $$\triangle A B C$$ with $$A E$$ as a side, point $$M$$ is the midpoint of $$E G$$, connect $$D M$$, connect $$B F$$, construct an equilateral triangle $$\triangle B F N$$ on the right side of $$B F$$, connect $$C N$$, connect $$M N$$, then the minimum value of $$C N + M N + D M$$ is          .

Answer: $$\frac{9}{2}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 8 \text{cm}$$, $$B C = 14 \text{cm}$$. Point P starts from point A and moves along the path $$A \rightarrow C \rightarrow B$$ toward the endpoint B. Point Q starts from point B and moves along the path $$B \rightarrow C \rightarrow A$$ toward the endpoint A. Points P and Q begin moving simultaneously at speeds of $$2 \text{cm} / \text{s}$$ and $$3 \text{cm} / \text{s}$$, respectively, and both must reach their respective endpoints before stopping. Perpendiculars are drawn from P and Q to line $$l$$, meeting it at points E and F, respectively. Let the time of motion be $$t$$ seconds. To make triangle $$P E C$$ congruent to triangle $$Q F C$$, the value of $$t$$ is        .

Answer: $$\frac{22}{5}$$ or 6 or 8


As shown in the figure, in the rhombus $$A B C D$$, $$\angle A = 60 \circ$$, $$G$$ is the midpoint of $$A D$$, point $$E$$ lies on the extension of $$B C$$, $$F$$ and $$H$$ are the midpoints of $$C E$$ and $$G E$$ respectively, $$\angle E H F = \angle D G E$$, $$C F = \sqrt{7}$$, then $$A B =$$     .

Answer: 4


$$\triangle A B C$$ is an isosceles right triangle, $$A B = A C = 6$$, $$\angle B A C = 90 \circ$$, point $$D$$ moves along side $$B C$$. Construct an isosceles right triangle $$\triangle A D E$$ with right angle at $$A$$ on the right side of $$A D$$ (as shown in the figure). $$M$$ is the midpoint of $$D E$$, $$N$$ is a trisection point of $$B C$$, $$C N = \frac{1}{3} B C$$, connect $$M N$$, then the minimum value of segment $$M N$$ is      .

Answer: 1


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, $$A B = 3$$, $$B C = 4$$, $$D$$ is the midpoint of side $$A C$$, connect $$B D$$, fold $$\triangle A B D$$ along $$B D$$ to obtain $$\triangle E B D$$, connect $$C E$$, then the distance from point $$E$$ to $$B C$$ is      .

Answer: $$\frac{21}{25}$$/$$0 . 84$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 4$$, M is a moving point on side $$A B$$ (excluding endpoints). Fold $$\triangle A D M$$ along the straight line $$D M$$ to obtain $$\triangle N D M$$; when ray $$C N$$ intersects segment $$A B$$ at point P, connect $$D P$$, then the maximum value of $$D P$$ is      .

Answer: $$2 \sqrt{5}$$


As shown in the figure, in rectangle $$A B C D$$, point $$E$$ lies on side $$A B$$, with $$B E = B C$$; point $$F$$ lies on side $$C D$$; connect $$E F$$, and fold quadrilateral $$B E F C$$ along $$E F$$, such that point $$B$$ lands exactly at point $$G$$ on side $$A D$$, and the image of point $$C$$ is point $$H$$. Given that $$A B = 3 A G = 9$$, find the length of $$E F$$.

Answer: $$\frac{5 \sqrt{10}}{3}$$


As shown in the figure, point $$D$$ is the midpoint of side $$AB$$ of $$\triangle ABC$$. Folding $$\triangle BCD$$ along the line $$CD$$ makes it coincide with $$\triangle ECD$$. If $$AB = 4$$, $$CD = 2$$, $$AE = 1$$, then the distance from point $$C$$ to line $$AB$$ is      .

Answer: $$\frac{\sqrt{15}}{2}$$/$$\frac{1}{2} \sqrt{15}$$


As shown in the figure, $$AC$$ is the diameter of circle O, $$AC = 4$$, $$\angle ACB = 60^\circ$$, point D is a moving point on chord $$AB$$, then the minimum value of $$OD + \frac{1}{2} BD$$ is           .

Answer: $$\sqrt{3}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle A B C = \angle A D C = 45^{\circ}$$, $$\angle B A D = \angle B C A = 105^{\circ}$$, if $$B C = \sqrt{6} - \sqrt{2}$$, then the length of $$C D$$ is        .

Answer: 2


As shown in the figure, $$AB$$ is the diameter of semicircle $$O$$, $$M$$ and $$C$$ are trisection points of the semicircle, $$AB = 8$$, $$BD$$ is tangent to semicircle $$O$$ at point $$B$$, point $$P$$ is a moving point on $$\overset{⌢}{AM}$$ (not coinciding with points $$A$$ or $$M$$), line $$PC$$ intersects $$BD$$ at point $$D$$, $$BE \bot OC$$ at point $$E$$, and extending $$BE$$ intersects $$PC$$ at point $$F$$. Which of the following conclusions are correct? (Write all correct conclusion numbers)
① $$PB = PD$$; ② The length of $$\overset{⌢}{BC}$$ is $$\frac{4}{3} \pi$$; ③ $$\angle DBE = 45^\circ$$; ④ $$\triangle BCF \sim \triangle PFB$$; ⑤ $$CF \cdot CP$$ is constant.

Answer: ②⑤


As shown in the figure, it is given that $$AB = 2 \sqrt{2}$$, and C is a moving point on segment $$AB$$. On the same side of $$AB$$, construct rhombuses $$ACED$$ and $$CBGF$$ with sides $$AC$$ and $$CB$$, respectively. Points C, E, F lie on a straight line, and $$\angle D = 120^\circ$$. P and Q are the midpoints of diagonals $$AE$$ and $$BF$$, respectively. When point C moves along segment $$AB$$, the shortest distance between points P and Q is     (answer in radical form).

Answer: $$\frac{\sqrt{6}}{2}$$


As shown in the figure, inside the square $ABCD$ with side length $10$ (excluding the boundary) there is a point $E$. Connect $CE$. From point $A$, construct $\angle BAF = \angle DCE$, and $AF = CE$. Connect $EF$. Rotate segment $EF$ clockwise around point $E$ by $90^\circ$, and point $F$ happens to fall on point $D$. Find the length of $EC$.

Answer: $$\text{2} \sqrt{\text{5}}$$


As shown in the figure, ∠MAN = 90°, point C lies on side AM, AC = 4, point B is a moving point on side AN, connect BC, △A′BC is symmetric to △ABC with respect to the line containing BC, points D and E are the midpoints of AC and BC respectively, connect DE and extend it to intersect the line containing A′B at point F, connect A′E. When △A′EF is a right triangle, the length of AB is     .

Answer: $$4 \sqrt{3}$$ or 4


As shown in the figure, in square $ABCD$ with side length $6$, points $E$ and $F$ are moving points on sides $AB$ and $BC$ respectively, satisfying $AE = BF$. $AF$ intersects $DE$ at point $O$. Point $M$ is the midpoint of $DF$, and point $G$ is on side $AB$ such that $AG = 2GB$. Find the minimum value of $OM + \frac{1}{2}FG$.

Answer: $$5$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, the angle bisector of $$\angle B A D$$ intersects $$B C$$ at point $$O$$. With $$O$$ as the center and $$O A$$ as the radius, an arc is drawn, which passes exactly through point $$D$$. The area of the shaded region in the figure is      .

Answer: $$2 \pi - \text{4}$$


As shown in the figure, the side length of square $$A B C D$$ is 8, and M and N are moving points on sides $$A B$$ and $$B C$$, respectively. An isosceles right triangle $$\text{Rt} \triangle M P N$$ is constructed with $$M N$$ as the hypotenuse (where $$M P = N P$$, $$\angle M P N = 90^\circ$$). Point E lies on side $$C D$$ such that $$D E = 3$$. Connect $$P E$$ and $$P C$$. Find the minimum perimeter of $$\triangle C P E$$      .

Answer: $$5 + \sqrt{73}$$/$$\sqrt{73} + 5$$


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, and D is a point in the plane. Connect $$A D$$, and rotate $$A D$$ clockwise around point D by $$60^\circ$$ to obtain segment $$D E$$. Connect $$B D$$ and $$C E$$. When $$\angle D A C = 30^\circ$$, $$A B = 2 \sqrt{3}$$, and $$A D = 4$$, then $$C E =$$          .

Answer: 2 or $$2 \sqrt{7}$$


As shown in the figure, quadrilateral $$ABCD$$ is a square, and point $$P$$ is a moving point on the circle $$\bigodot O$$ with diameter $$AD$$. Connect $$BP$$, and construct an equilateral triangle $$BPQ$$ with $$BP$$ as a side. Connect $$OQ$$. If $$AB = 2$$, then the maximum value of segment $$OQ$$ is        .

Answer: $$\sqrt{5} + 1$$/$$1 + \sqrt{5}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 60 \circ$$, $$B C = 2$$, $$D$$ is the midpoint of $$A B$$, and $$E$$ is a point on $$A C$$. If $$D E$$ bisects the perimeter of $$\triangle A B C$$, then the length of $$D E$$ is     .

Answer: $$\sqrt{3}$$


As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$A B = 6$$, $$\angle D A C = 60 \circ$$, point $$F$$ lies on segment $$A O$$, moving from point $$A$$ to point $$O$$, connect $$D F$$, and construct an equilateral triangle $$\triangle D F E$$ with $$D F$$ as a side, with point $$E$$ and point $$A$$ located on opposite sides of $$D F$$.

(1) When point $$F$$ moves to point $$O$$, the length of $$C E$$ is      ; (2) As point $$F$$ moves along segment $$A O$$ from point $$A$$ to point $$O$$, the minimum value of $$C E$$ is      .

Answer: $$6$$     $$3 \sqrt{3}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = A D = 6$$, point P moves inside $$\square A B C D$$, connecting $$P A$$, $$P B$$, $$P C$$. If $$\angle A P B = \angle A B C = 60^\circ$$, then the maximum value of $$P A + P C$$ is           .

Answer: $$4 \sqrt{3}$$


As shown in the figure, in $$\text{Rt} \Delta A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 6$$, $$B C = 8$$, D and E are two moving points on sides $$B C$$ and $$A C$$ respectively, with $$D E = 4$$, P is the midpoint of $$D E$$, connect $$P A$$ and $$P B$$, then the minimum value of $$P A + \frac{1}{4} P B$$ is   .

Answer: $$\frac{\sqrt{145}}{2}$$


As shown in the figure, the side length of square $$ABCD$$ is $$\sqrt{2}$$, and points E and G are the midpoints of sides $$AD$$ and $$BC$$, respectively. Connect $$BE$$, and fold $$\triangle ABE$$ along $$BE$$ into the same plane to obtain $$\triangle FBE$$. There is a point H on side $$CD$$; connect $$GH$$, and fold $$\triangle CGH$$ along $$GH$$ into the same plane to obtain $$\triangle PGH$$. If point P happens to fall on $$BF$$, then the length of the fold line $$GH$$ is       .

Answer: $$\frac{5 \sqrt{2}}{8}$$


As shown in the figure, in right triangle ABC, ∠ABC = 90°, ∠ACB = 30°, hypotenuse AC = 4, and point P is a moving point inside the triangle. The minimum value of PA + PB + PC is      .

Answer: $$2 \sqrt{7}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$B C = 6$$, point $$E$$ lies on side $$B C$$, $$C E = 2$$. If points $$P$$ and $$Q$$ are two moving points on sides $$C D$$ and $$A B$$ respectively, and segment $$P Q$$ always satisfies being perpendicular to $$A E$$ with foot of perpendicular at $$F$$, then the minimum value of $$A P + Q E$$ is      .

Answer: $$5 \sqrt{5}$$


As shown in the figure, in square $$ABCD$$, $$AB = a$$, points E and F lie on diagonal $$BD$$, and $$\angle ECF = \angle ABD$$. After rotating $$\triangle BCE$$ about point C by a certain angle, we obtain $$\triangle DCG$$, and connect $$FG$$. The following conclusions are given:
① $$\angle FCG = \angle CDG$$; ② $$BE^{2} + DF^{2} = EF^{2}$$; ③ $$FC$$ bisects $$\angle BFG$$; ④ The area of $$\triangle CEF$$ equals $$\frac{1}{4} a^{2}$$;
Among these, the correct conclusions are      . (Fill in all the correct conclusion numbers)

Answer: ①②③


As shown in the figure, in a square grid with side length 1, points A, B, C, D, E are all located at the vertices of the small squares. Connect AB and CD, intersecting at P. According to the auxiliary lines added as indicated in the figure, the value of cos∠BPC is        .

Answer: $$\frac{\sqrt{5}}{5}$$/$$\frac{1}{5} \sqrt{5}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ, \angle A = 60 \circ, A C = 1$$, rotate $$\triangle A B C$$ counterclockwise about point C to obtain $$\triangle A^{'} B^{'} C$$, at which point $$A^{'}$$ lies exactly on side $$A B$$. The distance between point $$B^{'}$$ and point B is

Answer: $$\sqrt{3}$$


As shown in the figure, $$l_{1}, l_{2}, l_{3}, l_{4}$$ are four parallel lines in the same plane, and the distance between any two adjacent parallel lines is $$h$$. The vertices of a square $$A B C D$$ with area 25 lie on these four lines respectively. What is the value of $$h$$?

Answer: $$\sqrt{5}$$


As shown in the figure, in the rhombus paper $$A B C D$$, $$A B = 1$$, $$\angle B = 60^\circ$$. The rhombus paper is folded along the crease $$E F$$, such that point $$D$$ lands at the midpoint $$G$$ of $$A B$$. Then the length of $$D E$$ is      .

Answer: $$\frac{7}{10}$$/$$0 . 7$$


The Huayang Peak is located in the Huayang Peak Park in Hengyang, and it is the foremost among the seventy-two peaks of Mount Heng. Wang Anshi once wrote a couplet: "Ten thousand miles away, the geese from Hengyang, regularly return here." In front of the peak, the Huayang Peak Square has been developed, with a large goose sculpture at its center, serving as the city emblem of Hengyang. A certain extracurricular practice group measured the height of the goose sculpture using a theodolite and a tape measure, obtaining the following data: as shown in the figure, $$A E = 10 \text{m}$$, $$\angle B D G = 30^\circ$$, $$\angle B F G = 60^\circ$$. It is known that the height of the theodolite $$D A$$ is $$1.5 \text{m}$$. Then the height of the goose sculpture $$B C$$ is approximately         $$\text{m}$$. (Round the result to $$0.1 \text{m}$$. Reference data: $$\sqrt{3} \approx 1.732$$)

Answer: 10.2


As shown in the figure, in rectangle ABCD, BD is a diagonal, AE ⊥ BD, with foot at E, connect CE. If $$\tan \angle ADB = \frac{1}{2}$$, then the value of $$\tan \angle DEC$$ is        .

Answer: $$\frac{2}{3}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, D and E are points on $$A B$$ and $$A C$$ respectively. Fold $$\triangle B C D$$ and $$\triangle A D E$$ along $$C D$$ and $$D E$$ respectively, such that points A and B coincide exactly at point $$A^{'}$$. Then $$\angle E A^{'} C =$$        °. If $$B C = 3$$, $$A C = 5$$, then $$A E =$$

Answer: $$90 \circ$$     $$\frac{8}{5}$$


As shown in the figure, in sector $$A O B$$, $$\angle A O B = 90 \circ$$, $$O A = 2$$, point $$C$$ is a point on $$O B$$. The sector $$A O B$$ is folded along $$A C$$, such that the image of point $$B$$, denoted $$B'$$, lies on ray $$A O$$. The area of the shaded region in the figure is         .

Answer: π+4–4$$\sqrt{2}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 4 \sqrt{3}$$, an arc is drawn with center at point B and radius AB, intersecting AC at point E and BC at point F. The area of the shaded region in the figure is   .

Answer: $$\frac{4 \pi}{3}$$/$$\frac{4}{3} \pi$$


As shown in the figure, the side length of square OABC is 2. Rotate square OABC counterclockwise about point O by an angle α (0° < α < 180°) to obtain square OA′B′C′. Connect BC′. When point A′ lies exactly on segment BC′, the length of segment BC′ is    .

Answer: $$\sqrt{6} + \sqrt{2}$$


As shown in the figure, in rectangle $$A B C D$$, E is the midpoint of side $$A B$$, and connecting $$D E$$ intersects diagonal $$A C$$ at point F. If $$A B = 8$$, $$A D = 6$$, then the length of $$A F$$ is      .

Answer: $$\frac{10}{3}$$/$$3 \frac{1}{3}$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, $$CD$$ is a chord of $$\bigodot O$$, connect $$AC$$, $$AD$$, if $$\angle ACD = 65^\circ$$, then the measure of $$\angle BAD$$ is       .

Answer: $$25 \circ$$


As shown in the figure, in rectangle $$A B C D$$, point $$E$$ is the midpoint of side $$A B$$. Fold $$\triangle A D E$$ along $$D E$$ to obtain $$\triangle F D E$$. Extend $$E F$$ to intersect $$B C$$ at point $$G$$, and connect $$B F$$ and $$D G$$.

(1) If $$\angle E D F = \alpha$$, then the measure of $$\angle E B F$$ is      . (Express in terms of $$\alpha$$) (2) If $$A B : A D = 2 : 3$$, then the tangent of $$\angle G E B$$ is      .

Answer: $$90 \circ - \alpha$$     $$\frac{3}{4}$$/$$0.75$$


As shown in the figure, in the paper of right triangle $$\text{Rt} \triangle A B C$$, ∠C = 90°, AC = 7, AB = 25, point D lies on side BC. Triangle △ADB is folded along AD to obtain triangle △ADB′, and AB′ intersects side BC at point E. If $$\triangle E D B^{'}$$ is a right triangle, then the length of BD is     .

Answer: 17 or $$\frac{75}{4}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$A D = 5, A B = 12, s i n A = \frac{4}{5}$$. Draw $$D E \bot A B$$, with foot at E, then $$s i n \angle B C E =$$      .

Answer: $$\frac{9 \sqrt{10}}{50}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C, \angle B A C = 90 \circ$$.Line l passes through point A. From point B, draw $$B E \bot l$$ meeting it at point E; from point C, draw $$C F \bot l$$ meeting it at point F. If $$B E = 2, C F = 5$$, then $$E F =$$          .

Answer: 7


As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 4$$, M is a moving point on side $$A B$$ (excluding endpoints). Fold $$\triangle A D M$$ along the straight line $$D M$$ to obtain $$\triangle N D M$$. When ray $$C N$$ intersects segment $$A B$$ at point P, connect $$D P$$. Then the area of $$\triangle C D P$$ is           ; the maximum value of $$D P$$ is           .

Answer: $$10$$     $$2 \sqrt{5}$$


$$\triangle D E F$$ is an equilateral triangle. Extend $$F D$$, $$D E$$, $$E F$$ to points $$A$$, $$B$$, $$C$$ respectively, such that $$D A = E B = F C$$. Connect $$A B$$, $$A C$$, $$B C$$. Connect $$B F$$ and extend it to intersect $$A C$$ at point $$G$$. If $$A D = D F = 2$$, then $$\angle D B F = \underset{\underline}{}$$   , $$F G = \underset{\underline}{}$$      .

Answer: $$30 \circ$$     $$\frac{4}{5} \sqrt{3}$$/$$\frac{4 \sqrt{3}}{5}$$


As shown in the figure, in $$\bigodot O$$, points A and B lie on $$\bigodot O$$, $$\angle A O B = 90 \circ$$, $$O A = 6$$, point C lies on OA such that $$O C = 2 A C$$, point D is the midpoint of OB, and point M is a moving point on the minor arc AB. Then the minimum value of $$C M + 2 D M$$ is   .

Answer: $$4 \sqrt{10}$$


As shown in the figure, points E and F are points on the sides of square $$A B C D$$, $$C E \bot D F$$, points G and H are the midpoints of segments $$C E$$ and $$D F$$ respectively, connect $$G H$$. If $$C F = 2$$, $$G H = 3 \sqrt{2}$$, then the length of $$A B$$ is             .

Answer: 8


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2, A D = \sqrt{7}$$, point $$P$$ moves along the sides of the rectangle in the order $$B \rightarrow C \rightarrow D \rightarrow A$$. When point $$P$$ does not coincide with points $$A$$ or $$B$$, fold $$\triangle A B P$$ along $$A P$$ to obtain $$\triangle A B^{'} P$$, and connect $$C B^{'}$$. Then, during the motion of point $$P$$, the minimum value of segment $$C B^{'}$$ is          .

Answer: $$\sqrt{11} - 2$$/$$- 2 + \sqrt{11}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle C = 35 \circ$$. Rotate $$\triangle A B C$$ around point A by $$\alpha \left(\right. 0 \circ < \alpha < 180 \circ \left.\right)$$, such that the rotated point B falls on $$B C$$, and the corresponding point of B is D. Connect $$A D$$, and $$A D$$ is the angle bisector of $$\angle B A C$$. Then $$\alpha =$$        .

Answer: $$\left(\frac{110}{3}\right) \circ$$


As shown in the figure, point E is a point outside square ABCD, ∠AEB = 90°. Rotate right triangle ABE counterclockwise around point A by 90° to obtain triangle ADF. Extend DF to intersect BE at point H. If BH = 7 and BC = 13, then DH =     .

Answer: 17


As shown in the figure, in $$\triangle A B C$$, $$AB = 4$$, $$AC = 6$$, $$E$$ is the midpoint of $$BC$$, $$AD$$ is the angle bisector of $$\triangle ABC$$, the area of $$\triangle ABC$$ is denoted as $$S_{1}$$, and the area of $$\triangle ADE$$ is denoted as $$S_{2}$$, then $$S_{2} : S_{1} =$$      .

Answer: $$1 : 10$$


As shown in the figure, construct a regular pentagon $ABCDE$ and an equilateral triangle $ABF$ on the same side of $AB$. Connect $FE$ and $FC$. Then the measure of $\angle EFA$ is            .

Answer: 66°


As shown in the figure, in $$\triangle A B C$$, $$A B = \sqrt{2}$$, rotate $$\triangle A B C$$ counterclockwise around point B by $$45 \circ$$ to obtain $$\triangle A_{1} B C_{1}$$, then the area of the shaded region is      .

Answer: $$\frac{\sqrt{2}}{2}$$


As shown in the figure, the rhombus $$A B C D$$ has a side length of $$6 \text{cm}$$, $$\angle B A D = 60 \circ$$, and when the rhombus is translated along the direction of AC by $$2 \sqrt{3} \text{cm}$$, it forms quadrilateral $$A ' B ' C ' D '$$, and $$A ' D '$$ intersects CD at point E. The distance from point E to AC is            $$\text{cm}$$.

Answer: 2


As shown in the figure, in square $$ABCD$$ with side length 4, point $$E$$ is a point on diagonal $$BD$$. If the circumcircle $$\bigodot O$$ of $$\triangle ABE$$ is tangent to side $$CD$$, then the radius of $$\bigodot O$$ is          .

Answer: $$\frac{5}{2}$$


As shown in the figure, in rectangle $$A B C D$$, points G and E lie on sides $$B C$$ and $$D C$$ respectively. Connect $$A G$$, $$E G$$, and $$A E$$. Fold $$\triangle A B G$$ and $$\triangle E C G$$ along $$A G$$ and $$E G$$ respectively, such that points B and C coincide at the same point on $$A E$$, denoted as point F. If $$C E = 3$$ and $$C G = 4$$, then $$\sin \angle D A E =$$       .

Answer: $$\frac{7}{25}$$


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, $$O$$ is the circumcenter of $$\triangle A B C$$, and the circumradius is $$2 \sqrt{3}$$. Arcs are drawn with centers at $$A$$, $$B$$, and $$C$$, and radii $$A O$$, $$B O$$, and $$C O$$, intersecting the three sides of $$\triangle A B C$$ at points $$H$$, $$I$$, $$D$$, $$E$$, $$F$$, and $$G$$, respectively. The area of the shaded region is      .

Answer: $$6 \pi - 9 \sqrt{3}$$


As shown in the figure, in a grid diagram composed of 4 identical small squares, ∠1 + ∠2 + ∠3 =        degrees.

Answer: 135


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle A = 36 \circ$$, if $$B D$$ is the angle bisector of $$\triangle A B C$$, then the measure of $$\angle B D C$$ is     .

Answer: $$72 \circ$$


As shown in the figure, in parallelogram $$\square A B C D$$, an arc is drawn with center $$A$$ and radius $$A B$$, intersecting $$B C$$ at point $$E$$. Connect $$A E$$, $$D E$$, and $$A C$$. If $$A C \bot A B$$, $$A B = 6$$, $$A D = 10$$, then the length of $$D E$$ is      .

Answer: 8


As shown in the figure, in △ABC, the perpendicular bisector MN of BC intersects AB at point D, and CD bisects ∠ACB. If AD = 2 and BD = 3, then the length of AC is     .

Answer: $$\sqrt{10}$$


Given rectangle $$A B C D$$, point $$E$$ is on side $$A D$$, with $$D E < A E$$, connect $$B E$$, point $$G$$ is on side $$B C$$, connect $$E G$$, $$B E$$ bisects $$\angle A E G$$, if $$B G = 5 G C$$, $$D E = 2 C G$$, $$B E = 2 \sqrt{10}$$, then the area of $$\triangle A B E$$ is           .

Answer: $$4 \sqrt{6}$$


As shown in the figure, in the regular hexagon $$A B C D E F$$ with side length 2, point $$P$$ lies on $$A F$$. A beam of light emanates from point $$P$$, strikes point $$Q$$ on the mirror surface $$B C$$, reflects, and then hits point $$G$$ on $$D E$$. Given that $$P Q \parallel A B$$ and $$Q G \parallel C D$$, find $$P Q + Q G =$$           .

Answer: 6


As shown in the figure, in heptagon $$A B C D E F G$$, the extensions of $$A B$$ and $$E D$$ intersect at point O, and the sum of the exterior angles $$\angle 1$$, $$\angle 2$$, $$\angle 3$$, $$\angle 4$$ is $$220^\circ$$. Then the measure of $$\angle B O D$$ is       $$^\circ$$.

Answer: 40


As shown in the figure, in square $$ABCD$$, points $$E$$, $$F$$, and $$G$$ lie on sides $$AD$$, $$BC$$, and $$CD$$ respectively, and $$BG \bot EF$$, with foot of perpendicular at $$H$$. Among the following conclusions: ① $$H$$ is the midpoint of segment $$BG$$; ② $$\angle DEF = \angle BGC$$; ③ $$BG = EF$$; ④ $$AE + FC = DG$$, the correct conclusions are        .

Answer: ②③④


As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$. Point $$E$$ lies on segment $$A B$$. Connect $$D E$$ and $$C E$$. Fold $$\triangle D A E$$ along $$D E$$ such that point $$A$$ coincides exactly with point $$O$$. If $$A D = 2 \sqrt{3}$$, then the length of $$C E$$ is      .

Answer: $$2 \sqrt{7}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 8, A D = 6$$, points E and F are moving points on sides $$A D$$ and $$C D$$ respectively. Connect $$B E$$ and $$E F$$. Point G is the midpoint of $$B E$$, and point H is the midpoint of $$E F$$. Connect $$G H$$. What is the maximum value of $$G H$$?

Answer: 5


As shown in the figure, $$OA$$, $$OB$$ are radii of $$\bigodot O$$, point C lies on $$\bigodot O$$, $$\angle AOB = 30^\circ$$, $$\angle OBC = 40^\circ$$, then $$\angle OAC =$$      $$^\circ$$.

Answer: 25


As shown in the figure, in parallelogram $$\square A B C D$$, $$A D = 12$$, and the circle $$\bigodot O$$ with diameter $$A D$$ is tangent to $$B C$$ at point $$E$$. Connect $$O C$$. If $$O C = A B$$, then the perimeter of $$\square A B C D$$ is            .

Answer: $$24 + 6 \sqrt{5}$$


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E and F are the midpoints of sides $$AB$$ and $$BC$$ respectively, connect $$AF$$ and $$DE$$, G and H are the midpoints of $$DE$$ and $$AF$$ respectively, connect $$GH$$, then the length of $$GH$$ is

Answer: $$\sqrt{2}$$


As shown in the figure, a 30° triangle ruler and a straightedge are stacked together. If $$\angle 1 = 72^\circ$$, then the measure of $$\angle 2$$ is      .

Answer: $$132 \circ$$


As shown in the figure, quadrilateral $$A B C D$$ is a regular quadrilateral inscribed in $$\bigodot O$$. With points A and O as centers, arcs are drawn with a fixed radius greater than $$\frac{1}{2} O A$$; these arcs intersect at points M and N. Draw line MN, which intersects $$\bigodot O$$ at points E and F. If $$O A = 1$$, then the area of the shaded region bounded by $$\overset{⌢}{B E}$$, $$A E$$, and $$A B$$ is       .

Answer: $$\frac{\pi}{12} + \frac{\sqrt{3}}{4} - \frac{1}{2}$$


As shown in the figure, in rectangle $$ABCD$$, an arc with center $$C$$ and radius $$CD$$ intersects $$AB$$ at point $$E$$, and an arc with center $$C$$ and radius $$CB$$ intersects $$CD$$ at point $$F$$. Connect $$AC$$. If $$AC = \sqrt{5}$$ and $$BC = 1$$, then the area of the shaded region in the figure is      (express the result in terms of $$\pi$$)

Answer: $$\frac{\sqrt{3}}{2} + \frac{1}{12} \pi$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = 7$$, $$B C = 8$$, $$s i n B = \frac{4}{5}$$. Point $$P$$ lies on side $$A B$$, $$A P = 2$$, and a circle $$\bigodot P$$ is drawn with center $$P$$ and radius $$A P$$. Point $$Q$$ lies on side $$B C$$, and a circle $$\bigodot Q$$ is drawn with center $$Q$$ and radius $$C Q$$. If $$\bigodot P$$ and $$\bigodot Q$$ are externally tangent, then the length of $$C Q$$ is        .

Answer: $$\frac{37}{14}$$/$$2 \frac{9}{14}$$


In Rt△ABC, ∠C=90°, AD bisects ∠CAB, AC=6, BC=8, CD=       .

Answer: 3.


As shown in the figure, in the regular pentagon $$A B C D E$$, $$A D$$ and $$C E$$ intersect at point F, and connect $$B F$$. Then the measure of $$\angle C F B$$ is       .

Answer: $$54 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 3$$, $$B C = 4$$, point D lies on side $$A B$$, $$A D = A C$$, $$A E \bot C D$$, with foot at F, intersecting $$B C$$ at point E. Find the length of $$B E$$:

Answer: $$\frac{5}{2}$$


As shown in the figure, $$AD$$ is the diameter of $$\bigodot O$$, and $$\triangle ABC$$ is an inscribed triangle of $$\bigodot O$$. If $$\angle DAC = \angle ABC$$, and $$AC = 5$$, then $$AD =$$      .

Answer: $$5 \sqrt{2}$$


As shown in the figure, the cross-section of the high-speed railway tunnel is part of a circle with center O. The road surface AB = 24 meters, and the clear height CD = 18 meters. Then the length of OD is ______.

Answer: 5


As shown in the figure, quadrilateral ABCD is a rhombus, $$\angle ABC = 70^\circ$$, extend BC to E, and within $$\angle DCE$$, draw ray CM such that $$\angle ECM = 15^\circ$$. From point D, draw $$DF \bot CM$$, with foot at F. If $$DF = \sqrt{6}$$, then the length of diagonal BD is      .

Answer: $$2 \sqrt{6}$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, point E is the midpoint of $$OB$$, and a chord $$CD \bot AB$$ is drawn through point E. Connect $$AC$$ and $$AD$$. If point F is the midpoint of $$\overset{⌢}{AC}$$, and a perpendicular $$CG \bot AF$$ is drawn from point C with foot at point G. If the radius of $$\bigodot O$$ is 2, then the length of $$CG$$ is        .

Answer: $$\sqrt{3}$$


As shown in the figure, in rectangle $$A B C D$$, point E lies on side $$B C$$, and point F lies on side $$A B$$. Fold $$C D$$ along $$D E$$ so that point C lands at point $$C^{'}$$, with $$D E$$ as the fold line; then fold $$B E$$ along $$E F$$ so that point B lands exactly at point $$B^{'}$$ on segment $$E C^{'}$$, with $$E F$$ as the fold line. If $$C D = 8$$, $$B F = 3$$, and $$B^{'} C^{'} = 2$$, then the length of $$B C$$ is      .

Answer: 10


As shown in the figure, given that line $$a \parallel b$$, $$\angle 1 = 85^\circ$$, $$\angle 2 = 60^\circ$$, then $$\angle 3 =$$ .

Answer: $$35 \circ$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$B C = 4$$, $$A C$$ is sufficiently long, points D and E lie on sides $$B C$$ and $$A C$$ respectively, F is the midpoint of $$D E$$, and if $$B D = C E$$, then the minimum length of $$C F$$ is         .

Answer: $$\sqrt[]{2}$$


As shown in the figure, $$AB$$ is a chord of $$\bigodot O$$, $$BC$$ is a line passing through point $$B$$, $$\angle AOB = 130^\circ$$, when $$\angle ABC =$$      , $$BC$$ is a tangent to $$\bigodot O$$.

Answer: $$65 \circ$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, $$B C = 2 \sqrt{2}$$, an arc is drawn with center at point A and radius $$A D$$, intersecting $$B C$$ at point E. Connect $$A E$$. The area of the shaded region in the figure is      .

Answer: $$4 \sqrt{2} - 2 - \pi$$


As shown in the figure, given rectangle $$A B C D$$, with $$A B = 4$$, $$A D = 3$$, point E is a moving point on side $$D C$$, not coinciding with either endpoint. Connect $$B E$$, and fold $$B C E$$ along $$B E$$ to obtain $$B E F$$. Connect $$A F$$ and extend it to intersect $$C D$$ at point G. Then the maximum value of segment $$C G$$ is       .

Answer: $$4 - \sqrt{7}$$/$$- \sqrt{7} + 4$$


As shown in the figure, in the rhombus $$ABCD$$ with side length 8, $$E$$ is the midpoint of side $$AD$$, and connecting $$CE$$ intersects diagonal $$BD$$ at point $$F$$. If $$\angle DEF = \angle DFE$$, then the area of this rhombus is        .

Answer: $$24 \sqrt{7}$$


As shown in the figure, a cylindrical glass cup is placed horizontally, with a cross-section that is a circle of radius $$5 \text{cm}$$. The width of the water surface inside the cup is $$AB = 8 \text{cm}$$. The depth of the water $$CD$$ is           .

Answer: $$2 ( \text{cm} )$$