As shown in the figure, quadrilateral $$ABCD$$ is a square, $$\triangle DEF$$ rotates about point $$D$$ ($$DE < AB$$), $$\angle EDF = 90^\circ$$, $$DE = DF$$, connect $$AE$$, $$CF$$; line $$AE$$ intersects line $$CF$$ at point $$G$$. Connect $$BG$$. If $$AB = 6$$, $$DE = 2\sqrt{2}$$, write down, during the rotation of $$\triangle DEF$$,

① When point $$E$$ is inside square $$ABCD$$ and $$EF \perp CD$$, $$BG =$$      ; ② The minimum length of segment $$BG$$      .

Answer: $$\frac{12 \sqrt{10}}{5}$$/$$\frac{12}{5} \sqrt{10}$$     $$2 \sqrt{14}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ is a diagonal, $$A B = C D$$, $$\angle D A C + \angle B C A = 180 \circ$$, $$\angle B A C + \angle A C D = 90 \circ$$. If the area of quadrilateral $$A B C D$$ is 7, then the length of $$C D$$ is         .

Answer: $$\sqrt{14}$$


As shown in the figure, in $$\triangle A B C$$, segment AD is a median, point O is the midpoint of segment AD, line l passes through point O, and points B and C lie on the same side of l. Perpendiculars are drawn from points B, C, D, and A to line l, with feet of the perpendiculars being points E, F, H, and G, respectively. Which of the following statements are always correct?

① $$\triangle A I G \approx \triangle B I E$$; ② $$A G = D H$$; ③ $$2 A G = B E + C F$$; ④ If points B and C lie on opposite sides of l, then $$2 A G = B E - C F$$.

Answer: ②③/③②


As shown in the figure, the diagonals $$AC, BD$$ of square $$ABCD$$ intersect at point $$O$$, point $$F$$ is a point on $$CD$$, $$OE \bot OF$$ intersects $$BC$$ at point $$E$$, connect $$AE, BF$$ intersecting at point $$P$$, connect $$OP$$. The following conclusions:

① $$AE \bot BF$$; ② $$\angle OPA = 45^\circ$$; ③ $$AP - BP = \sqrt{2} OP$$; ④ If $$BE : CE = 2 : 3$$, then $$\tan \angle CAE = \frac{4}{7}$$; ⑤ The area of quadrilateral $$OECF$$ is $$\frac{1}{4}$$ of the area of square $$ABCD$$. Among these, the correct conclusions are         .

Answer: ①②③⑤


As shown in the figure, in $$R t \triangle A B C$$, $$\angle C = 90 \circ$$, $$A C = 2$$, $$B C = 4$$, points E and F lie on $$A B$$ and $$A C$$ respectively, and the reflection of point A over line EF, denoted $$A^{'}$$, lies on $$B C$$. Let $$C A^{'} = x$$. If $$A E = A F$$, then x =       ; let $$A E = y$$, write the function expression of y in terms of x:           .

Answer: $$\sqrt{5} - 1$$     $$y = \frac{\sqrt{5} x^{2} + 4 \sqrt{5}}{4 x + 4}$$


As shown in the figure, in rhombus $$A B C D$$, $$\angle A = 60^\circ$$, $$A D = 4$$, point M is a point on $$A D$$ such that $$A M = 3$$, and point P is a moving point on $$A B$$. Fold $$\triangle A M P$$ along $$M P$$ to obtain $$\triangle Q M P$$, where point A corresponds to point Q. Connect $$D Q$$. When point Q lies on a diagonal of rhombus $$A B C D$$, the length of $$D Q$$ is      .

Answer: $$\frac{1 + \sqrt{33}}{2}$$ or $$\sqrt{7}$$


As shown in the figure: In equilateral triangle ABC, $$AB = 1$$, E and F are moving points on sides $$AB$$ and $$AC$$ respectively, and $$CF = 2BE$$. Find the minimum value of $$BF + 2CE$$          .

Answer: $$\sqrt{7}$$


As shown in the figure, the side length of square $$A B C D$$ is 2, M is the midpoint of $$B C$$, and N is a moving point on $$A M$$. Through point N, draw $$E F \bot A M$$, intersecting $$A B$$ and $$C D$$ at points E and F, respectively. If the value of $$E M + A F$$ is $$\sqrt{10}$$, then the length of $$D F$$ is      .

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


As shown in the figure, in $$\triangle A B C$$, point D is a point inside $$\triangle A B C$$, $$B D$$ bisects $$\angle A B C$$, $$C D \bot B D$$ at point D, $$\angle A C D = \angle A$$. If $$c o s \angle D C B = \frac{1}{3}$$, $$A B = 5$$, then the length of $$B C$$ is          .

Answer: 3


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = \sqrt{2}$$, $$B C = \sqrt{6}$$, $$\angle A B C = 60^\circ$$. Construct squares outwardly on each side of $$\square A B C D$$, and connect the centers of these four squares in order to form quadrilateral $$E F G H$$. The area of quadrilateral $$E F G H$$ is      .

Answer: $$7$$


As shown in the figure, in rhombus $$A B C D$$, $$\angle B = 60 \circ$$, E is a point on side $$A B$$, triangle $$\triangle B C E$$ is folded along $$C E$$, and the image of point B is point F, $$C F$$ intersects $$A D$$ at G, $$E F$$ intersects $$A D$$ at H. If $$A B = 3 A E$$, then $$C G : F G =$$     .

Answer: $$7 : 1$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C$$, $$\angle A B C = 45 \circ$$, point $$D$$ is the midpoint of $$B C$$, $$C E \bot A D$$ at point $$E$$, and its extension intersects $$A B$$ at point $$F$$, $$B M \bot C F$$ intersects the extension of $$C F$$ at point $$M$$, $$C N$$ bisects $$\angle A C B$$ and intersects $$A D$$ at point $$G$$, and intersects $$A B$$ at point $$N$$, connect $$D F$$, then the following conclusions: ① $$\angle A D C = \angle B D F$$; ② $$A D = C F + D F$$; ③ $$S_{\triangle A C D} = 3 S_{\triangle C D F}$$; ④ $$\angle A G N = \angle B F D$$; ⑤ $$A E - B M = E M$$; ⑥ $$\angle A D F = 2 \angle C A D$$. Among these, the correct ones are        . (Only fill in the serial numbers)

Answer: ①②③④⑤⑥


As shown in the figure, it is given that the diagonals $$AC$$ and $$BD$$ of rectangle $$ABCD$$ intersect at point O, and point E is a moving point on side $$AB$$, with $$CE$$ intersecting $$BD$$ at point F, and connecting $$OE$$.

(1) If point E is the midpoint of $$AB$$, then $$\frac{OF}{FB}$$ =       ; (2) If point F is the midpoint of $$OB$$, then $$\frac{AE}{BE}$$ =         .

Answer: $$\frac{1}{2}$$/0.5     2


As shown in the figure, point E is inside $$\square A B C D$$, EB⊥BC, ED⊥CD, and $$\angle E A B = 45 \circ$$. Connect CE. For the following four conclusions: ① $$\angle A D E = \angle A B E$$; ② $$\angle D A E = \angle D C E$$; ③ $$B E = A D$$; ④ When $$\angle D A B = 60 \circ$$, $$C E = 2 A E$$, the sequence numbers of all correct conclusions are        .

Answer: ①②③④


As shown in the figure, in the equilateral triangle $$\triangle ABC$$, $$AB = BC = AC = 5$$, point $$M$$ is a point on the line containing the altitude $$AD$$ from side $$BC$$, and an equilateral triangle $$\triangle BMN$$ is constructed with $$BM$$ as a side. Connect $$DN$$; then the minimum value of $$DN$$ is      .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 4 \sqrt{2}$$, E is the midpoint of $$A B$$, and F is a moving point on side $$A D$$ (point F does not coincide with points A or D). Fold $$\triangle A E F$$ along the line $$E F$$, and let the image of point A be $$A^{'}$$. Connect $$A^{'} D$$ and $$A^{'} C$$. When $$\triangle A^{'} D C$$ is an isosceles triangle, the length of $$A F$$ is       .

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


As shown in the figure, fold the paper triangle △ABC along MN such that point C lands inside quadrilateral ABNM; then there is a constant quantitative relationship among ∠1, ∠2, and ∠C. This relationship is   .

Answer: 2∠C=∠1+∠2


As shown in the figure, in △ABC, ∠BAC = 120°, AB = AC, point D is a point on side BC, point E is a point on segment CD, and CE = 1, AB = 2$$\sqrt{3}$$, ∠DAE = 60°, then the length of DE is         .

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


As shown in the figure, AB is the diameter of ⊙O, C is a point on the circle, and ∠AOC = 120°. The radius of ⊙O is 2. P is a moving point on the circle, and Q is the midpoint of AP. Then the maximum and minimum values of the length of CQ are     .

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


As shown in the figure, in rectangle ABCD, AB = 4, BC = $$4 \sqrt{3}$$, point P moves along segment BC (including points B and C). Connect AP. Rotate segment AP counterclockwise 60° about point A to obtain AQ, and connect DQ. What is the minimum value of segment DQ?

Answer: 2


As shown in the figure, AB is the diameter of ⊙O, and side DE of rectangle ACDE is tangent to ⊙O, with point C on ⊙O. If $$\frac{A E}{D E} = \frac{1}{4}$$, $$B D = \frac{8}{3}$$, then $$A B =$$     .

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


As shown in the figure, in square $$A B C D$$, points E and F lie on sides $$A B$$ and $$B C$$ respectively, with $$A E = 2$$, $$F C = 5$$, and $$\angle E D F = \angle F D C$$. The side length of square $$A B C D$$ is      .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A = 90 \circ$$, $$A B = 6$$, $$A C = 3$$, $$D$$ is a point on side $$A B$$ such that $$A D = 2 B D$$, draw $$D E \bot D C$$, intersecting $$B C$$ at point $$F$$, connect $$C E$$, if $$\angle D C E = \angle B$$, then the value of $$\frac{E F}{D F}$$ is      .

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


As shown in the figure, given ∠MON = 30°, B is a point on OM, BA ⊥ ON at A, quadrilateral ABCD is a square, P is a moving point on ray BM, connect CP, rotate CP 90° clockwise around point C to obtain CE, connect BE. If AB = $$\sqrt{3}$$, then the minimum value of BE is        .

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


As shown in the figure, in the right trapezoid $$A B C D$$, $$\angle A = \angle B = 90 \circ$$, the angle bisectors of $$\angle A D C$$ and $$\angle B C D$$ intersect exactly at point $$E$$ on side $$A B$$. Rotate $$\triangle C B E$$ counterclockwise about point $$E$$ to $$\triangle E F G$$, such that point $$B$$ lands at point $$F$$ on segment $$E C$$, and point $$C$$ lands at point $$G$$. Let $$E G$$ and $$F G$$ intersect $$C D$$ at points $$M$$ and $$N$$, respectively. Given $$A B = 2$$ and $$B C = 2 \sqrt{2}$$, find the value of $$D M : M N$$.

Answer: $$1 : 2$$


As shown in the figure, rectangle ABCD, point M is a point on the ray CD, AH ⊥ BM, with foot of perpendicular H. Fold △ABH along AB to obtain △ABH′. P and Q are the midpoints of AH′ and BH′, respectively. If AD = 2 and AB = 6, then the maximum value of PQ + CQ is              .

Answer: 7


As shown in the figure, in right triangle ABC, ∠BCA = 90°, AB = $$\sqrt{5}$$, AC = 2, D is a moving point on the hypotenuse AB (not coinciding with points A or B), DE ⊥ AC, DF ⊥ BC, with feet at E and F respectively. Connect EF; then the minimum value of EF is     .

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


As shown in the figure, if point $$O$$ is a point outside the square $$A B C D$$, $$O D = 2 \sqrt{2}$$, $$O C = 1$$, $$B O = \sqrt{10}$$, then the tangent of $$\angle C D O$$ is         .

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


As shown in the figure, given $$\triangle A B C$$, $$\triangle D C E$$, $$\triangle F E G$$ are three congruent isosceles triangles, with bases $$B C$$, $$C E$$, $$E G$$ lying on the same straight line, and $$A B = \sqrt{3}$$, $$B C = 1$$, $$A G$$ intersects $$D C$$, $$D E$$, $$E F$$ at points $$P$$, $$Q$$, $$R$$, respectively. Then $$P Q =$$      .

Answer: $$\frac{1}{2}$$/0.5


As shown in the figure, two paper strips of equal width are crossed and overlapped. If the quadrilateral $$A B C D$$ formed by the overlapping region has $$A B = 3$$ and $$A C = 2$$, then the length of $$B D$$ is               .

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


As shown in the figure, rotate the equilateral triangle $$\triangle A B C$$ clockwise around point C by 120° to obtain $$\triangle E D C$$. Draw $$A N \bot B C$$ intersecting $$B D$$ at F, and let point P be a point on $$C F$$. Connect $$A D$$, $$B D$$, $$C F$$, $$P A$$, and $$P B$$. The following conclusions hold:
① The four sides of quadrilateral $$A C E D$$ are equal; ② $$B P$$ bisects $$\angle D B C$$; ③ $$S_{\triangle B D E} = 12 S_{\triangle C N F}$$; ④ $$2 P B + P C > 6 F N$$. The correct conclusions are          .

Answer: ①③/③①


As shown in the figure, in rectangle $$A B C D$$, $$A B = 5$$, $$A D = 30$$, point E lies on $$A D$$ such that $$D E = 6$$. 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: 13


As shown in the figure, in parallelogram $$A B C D$$, $$A D = 2 A B$$, $$C E \bot A B$$ at point E, points F and G are the midpoints of $$A D$$ and $$B C$$ respectively, and connect $$C F$$, $$E F$$, $$F G$$. The following four statements: ① $$C E \bot F G$$; ② quadrilateral $$A B G F$$ is a rhombus; ③ $$B C = 2 E G$$; ④ $$\angle D F C = \angle E F G$$. The correct ones are         . (fill in the serial numbers)

Answer: ①②③④


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90^\circ$$, $$\angle B = 30^\circ$$, $$A C = 8$$. D is a moving point on $$B C$$, connect $$A D$$, the perpendicular bisector of $$A D$$ intersects $$A C$$ and $$A B$$ at points E and F, respectively. The maximum length of segment $$B F$$ is       .

Answer: $$\frac{32}{3}$$


As shown in the figure, point $$C$$ lies on segment $$AB$$, $$\triangle DAC$$ is an equilateral triangle, and quadrilateral $$CDEF$$ is a square. Point $$P$$ is a moving point on segment $$AE$$. Connect $$PB$$ and $$PC$$. If $$AC = 2$$ and $$BC = 3$$, then the minimum value of $$PB + PC$$ is      .

Answer: $$\sqrt{29}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 8, B C = 12$$, points $$E$$, $$F$$, $$G$$ move along sides $$A D$$, $$B C$$, $$A B$$ respectively, and segment $$E F$$ always passes through the center of symmetry of the rectangle. The minimum perimeter of $$\triangle E F G$$ is         .

Answer: $$4 \sqrt{13} + 8$$


In $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, points $$D$$, $$E$$ lie on side $$B C$$ (point $$D$$ is to the left of point $$E$$), $$\angle C A D = 2 \angle B A D$$, $$A D = C E$$, point $$F$$ lies on side $$A C$$, $$\angle B D A = \angle E F A$$, if $$C F = 4$$, $$C D = a$$, $$E B = b$$, then $$A D =$$      . (Express in terms of $$a$$, $$b$$)

Answer: $$a + 2 - b$$


As shown in the figure, the radius of ⊙O is 5, P is a point on the extension of diameter AB, BP = 1, CD is a chord of ⊙O with CD = 6. A parallelogram PCED is constructed with PC and PD as adjacent sides. When points C and D move on the circumference of the circle, the minimum length of segment PE is               .

Answer: 4


As shown in the figure, in rectangle ABCD, AD = 6, DC = 8, point H is on side AD, AH = 2, and E is a moving point on side AB. Connect HE. Construct a rhombus HEFG with HE as one side, located above and to the right of HE, such that point G lies on side DC. Connect CF.

(1) When rhombus HEFG is a square, the length of DG is ; (2) As point E moves, the range of values for the area S of triangle FCG is .

Answer: 2     $$8 - 2 \sqrt{13} \leq S \leq 8$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 1$$, $$B C = \sqrt{3}$$, point $$M$$ is a moving point on diagonal $$A C$$, connect $$D M$$, then the minimum value of $$D M + \frac{1}{2} A M$$ is      .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = B C$$, $$A D \bot B C$$, $$C E \bot A B$$, with feet at points $$D$$ and $$E$$ respectively. Point $$F$$ lies on the extension of $$D A$$, and connecting $$B F$$ intersects the extension of $$C E$$ at point $$M$$. $$A D = 2 C D$$.

(1) If $$A E = 2$$, then $$B D =$$  ;

(2) If $$B M : M F = 6 : 7$$, $$E M = 1$$, then $$A F =$$  .

Answer: 3     $$\frac{65}{18}$$/$$3 \frac{11}{18}$$


In $$\text{Rt} \Delta A B C$$, $$\angle A = 90 \circ$$, $$A B = \sqrt{5}$$, $$A C = 2 \sqrt{5}$$, $$M$$ is the midpoint of $$A C$$, and $$\Delta A B C$$ is rotated counterclockwise about point $$M$$ to obtain $$\Delta A^{'} B^{'} C^{'}$$. During the rotation, the lines $$B B^{'}$$ and $$C C^{'}$$ intersect at point $$P$$. Then the maximum value of $$B P$$ is           .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = a$$, $$B E = 6$$, $$E C = 10$$, fold $$\triangle A B E$$ along $$A E$$; if the image point F of point B falls exactly on the diagonal of rectangle $$A B C D$$, then the value of $$a$$ is               .

Answer: 12 or $$4 \sqrt{6}$$


As shown in the figure, $$AE$$ is the angle bisector of $$\angle CAM$$, point $$B$$ lies on ray $$AM$$, $$DE$$ is the perpendicular bisector of segment $$BC$$ intersecting $$AE$$ at $$E$$, and from point $$E$$, a perpendicular is drawn to $$AM$$ intersecting $$AM$$ at point $$F$$. If $$\angle ACB = 28^\circ$$ and $$\angle EBD = 25^\circ$$, then $$\angle AED =$$ $$^\circ$$.

Answer: $$37$$


As shown in the figure, in rhombus ABCD with side length 1, $$\angle ABC = 60^\circ$$. Translate $$\triangle ABD$$ along the direction of ray BD to obtain $$\triangle A'B'D'$$. Connect $$A'C$$, $$A'D$$, and $$B'C$$. The minimum value of $$A'C + B'C$$ is      .

Answer: $$\sqrt{3}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 4$$. Fold $$\triangle B C D$$ along diagonal $$B D$$ so that point C lands at $$C^{'}$$. Let $$B C^{'}$$ intersect $$A D$$ at point G. Points E and F lie on $$C^{'} D$$ and $$B D$$ respectively. Segment $$E F$$ intersects $$A D$$ at points H and M. Fold $$\triangle F D E$$ along $$E F$$ so that point D lands at point A. Let segment $$A E$$ intersect $$B C^{'}$$ at point M. Find the length of segment $$C^{'} M$$.

Answer: $$\frac{308}{527}$$


As shown in the figure, given $$\triangle A B C$$, $$A B = A C$$, $$B C = 6$$, $$\angle B A C = 120^\circ$$, point D lies on $$B C$$ (not coinciding with B or C), connect $$A D$$, and reflect $$\triangle A B D$$ and $$\triangle A C D$$ over lines $$A B$$ and $$A C$$ to obtain $$\triangle A B F$$ and $$\triangle A C E$$, respectively. Connect $$E F$$. The following conclusions are given: ① $$E F = \sqrt{3} A F$$; ② When $$A D \bot A F$$, the length of $$C D$$ is $$2 \sqrt{3}$$; ③ When points D, A, and F are collinear, quadrilateral $$A D C E$$ is a rhombus; ④ The minimum area of $$\triangle A E F$$ is $$\frac{3 \sqrt{3}}{4}$$. The correct conclusions are ______. (Fill in the serial numbers)

Answer: ①②③④


As shown in the figure, in rhombus $$A B C D$$, $$A B = 9$$, $$\angle A B C = 60 \circ$$, point $$E$$ lies on side $$A B$$ such that $$B E = 2 A E$$, and point $$P$$ moves along side $$B C$$. Connect $$P E$$, and rotate segment $$P E$$ clockwise by $$60 \circ$$ about point $$P$$ to obtain segment $$P F$$. Connect $$A F$$. What is the minimum length of segment $$A F$$?

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


As shown in the figure, in rectangle ABCD, AB = 12, BC = 16, AC and BD intersect at O, E is a point on DC, and from point O, draw OF ⊥ OE intersecting BC at F. Let d = $$\sqrt{DE^{2} + BF^{2}}$$, then the minimum value of d is      .

Answer: 10


As shown in the figure, $$\triangle A B C$$ is an isosceles right triangle, $$\angle A B C = 90 ^{\circ}$$, points $$D$$, $$E$$ move along sides $$A C$$, $$B C$$ respectively, connecting $$A E$$, $$B D$$ intersecting at point $$F$$, and always satisfying $$C E = \sqrt{2} A D$$. Which of the following conclusions are correct: ① $$\frac{B D}{A E} = \frac{\sqrt{2}}{2}$$; ② $$\angle D F E = 120^{\circ}$$; ③ $$\sqrt{2} B D \cdot B F = A C \cdot B E$$. The correct ones are                     .

Answer: ①


As shown in the figure, in rectangle ABCD, BD is the diagonal. Fold rectangle ABCD along the lines BE and BF such that point A lands at point M on BD, and point C lands at point N on BD. Connect EF. Given that AB = 6 and BC = 8, the length of EF is                  .

Answer: $$\frac{5 \sqrt{13}}{3}$$


As shown in the figure, the radius of sector $$A O B$$ is 8, $$\angle A O B = 90 \circ$$, points C, D, E are moving points on arc $$A B$$, radius $$O A$$, and $$O B$$ respectively, $$\angle C E D = 90 \circ$$, $$\text{tan} \angle C D E = \frac{2}{3}$$, then the minimum perimeter of $$\triangle C D E$$ is        .

Answer: $$\frac{40 \sqrt{13}}{13} + 8$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2 \sqrt{3}$$, point $$E$$ lies on side $$A D$$, $$E D = 2$$, an arc is drawn with center at point $$E$$ and radius equal to $$A E$$, intersecting $$B C$$ at point $$F$$, and passing exactly through point $$C$$. Connect $$A C$$ and $$C E$$. The area of the shaded region is      .

Answer: $$\frac{16 \pi}{3} - 4 \sqrt{3}$$


Given ∠AOB = 45°, point P is inside ∠AOB, point P1 is symmetric to point P with respect to OA, point P2 is symmetric to point P with respect to OB, connect P1P2 intersecting OA and OB at E and F respectively. If P1E = $$\frac{1}{2}$$ and OP = $$\sqrt{2}$$, then the length of EF is     .

Answer: $$\frac{5}{6}$$


As shown in the figure, in square $$ABCD$$, $$AB = 2$$, point $$E$$ is a point on diagonal $$AC$$ such that $$AE = CB$$. Connect $$DE$$ and extend it to intersect $$BC$$ at point $$G$$. From point $$A$$, draw $$AH \bot BE$$ meeting at point $$H$$, and intersect $$BC$$ at point $$F$$. Among the following conclusions: ① $$\triangle ABF \sim \triangle DCG$$; ② $$\angle BEG = 45^\circ$$; ③ $$4BH^2 = BG \cdot CD$$; ④ $$BF = 2\sqrt{2} - 1$$. The correct conclusions are         .

Answer: ①②③


As shown in the figure, the radius of semicircle O is 2, and E is a point on the semicircle. Fold point E onto diameter AB (with EE′ ⊥ AB). When the folded arc intersects diameter AB at least at one point, the range of possible lengths for the crease CD is                 .

Answer: $$2 \sqrt{3} \leq C D \leq 4$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 120 \circ$$, $$A C = 8$$, $$B C = 4$$. Fold side $$B C$$ along $$C E$$ such that point B lands at point D on $$A B$$, then fold side $$A C$$ along $$C F$$ such that point A lands at point $$A^{'}$$ on the extension of $$C D$$. The two creases intersect the hypotenuse $$A B$$ at points E and F, respectively. The length of segment $$F A^{'}$$ is        .

Answer: $$\frac{8 \sqrt{7}}{7}$$


As shown in the figure, BP bisects ∠ABC, $$\text{PD} \bot \text{BC}$$, and E, F are points on the two sides of the angle. There are four conclusions; those that can definitely lead to the remaining conclusions are: ① $$\angle \text{ABC} + \angle \text{EPF} = 180 \circ$$; ② $$\angle \text{BEP} = \angle \text{PFC}$$; ③ $$\text{PE} = \text{PF}$$; ④ $$2 \text{BD} = \text{BF} + \text{BE}$$,     .

Answer: From (1), (2), and (4), the other three can be derived.


△ABC is an equilateral triangle with side length 2, and point P is a moving point on line BC. Rotate segment AP counterclockwise 60° about point A to obtain AE. O is a moving point on side AB. Then the minimum value of OE is    .

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


As shown in the figure, in square $$ABCD$$, point $$E$$ lies on diagonal $$BD$$ such that $$BE = BC$$. Connect $$CE$$ and extend it to intersect $$AD$$ at point $$F$$. Connect $$AE$$, and from point $$B$$, draw $$BG \bot AE$$ intersecting at point $$G$$. Extend $$BG$$ to intersect $$AD$$ at point $$H$$. Among the following conclusions: ① $$AH = DF$$; ② $$\angle AEF = 45^\circ$$; ③ $$AH = DE$$; ④ $$S_{\text{quadrilateral } EFHG} = S_{\triangle DEF} + S_{\triangle AGH}$$, the correct conclusions are      $$\left(\right.$$fill in the correct serial numbers$$\left.\right)$$

Answer: ①②③


In the rhombus $$A B C D$$, $$\angle A = 120^\circ$$, $$A B = 4$$, there are two moving points E and F on sides $$A D$$ and $$C D$$ respectively, always satisfying $$D E = D F$$. Connect $$B E$$ and $$E F$$, take the midpoint G of $$B E$$ and connect $$F G$$. Then the minimum value of $$F G$$ is     .

Answer: 3


As shown in the figure, point O is a point inside triangle $$A B C$$, $$O A = O B = O C = 4, \angle B A C = 45 \circ$$, and it is given that $$S_{\triangle A O C} - S_{\triangle A O B} = 2$$, then $$\angle B O C =$$        , $$S_{\triangle A B C} =$$           .

Answer: $$90 \circ$$     $$2 \sqrt{31} + 8$$


As shown in the figure, $$\angle B A C =\angle D A F =\text{90}°$$, $$A B = A C$$, $$A D = A F$$, $$A B$$ and $$F E$$ intersect at point $$M$$, points $$D$$, $$E$$ are two points on side $$B C$$, and $$\angle D A E =\text{45}°$$, connect $$E F$$, $$B F$$, then the following conclusions: ① $$\triangle A F B ≌\triangle A D C$$; ② $$B E^{\text{2}} + D C^{\text{2}} = D E^{\text{2}}$$; ③ $$A B - A D = E D - B E$$; ④ only when $$\angle A M E =\text{90}°$$, $$B F = B E$$, among which the correct ones are    . (fill in the serial numbers)

Answer: ①②④


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$A B = 10 \text{cm}$$, $$A C = 8 \text{cm}$$, $$O$$ is the midpoint of $$A B$$, point $$P$$ starts from point $$C$$ and moves toward point $$B$$ along side $$C B$$ at a speed of $$1 \text{cm}/\text{s}$$, and point $$Q$$ starts from point $$A$$ and moves toward point $$C$$ along side $$A C$$ at a speed of $$2 \text{cm}/\text{s}$$. Points $$P$$ and $$Q$$ start simultaneously, and when one point reaches its endpoint, the other point also stops moving simultaneously.
(1) After 2 seconds of departure, the distance between points $$P$$ and $$Q$$ is                  $$\text{cm}$$.
(2) When the area of $$\triangle O P Q$$ is minimized, the distance between points $$P$$ and $$Q$$ is                     $$\text{cm}$$.

Answer: $$2 \sqrt{5}$$     $$\frac{\sqrt{61}}{2}$$


As shown in the figure, in quadrilateral ABCD, AB = AD = 2, ∠DAB = ∠DCA = 60°, then the maximum value of $$CD + CB$$ is          .

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


As shown in the figure, in quadrilateral ABCD, ∠A = 30°, AB = 8, point O is the midpoint of AB, OC = OD = $$\frac{1}{2}$$AB, OC ∥ AD, point P is a moving point on AB (not coinciding with points A or B), then the minimum value of PD + PC is      .

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


As shown in the figure, in △ABC, D is the midpoint of side BC. Connect AD, and fold △ACD along AD to obtain △ADF. DF intersects AB at point E. Connect BF, with BD = BF = 2 and AD = 3.

(1) Connect CF; the measure of ∠DCF is ______ degrees; (2) The distance from point D to AF is ______.

Answer: 30     $$\frac{3 \sqrt{21}}{7}$$


As shown in the figure, in $$\triangle A B C$$, $$A C = 5$$, $$B C = 4 \sqrt{3}$$, $$\angle A C B = 60 \circ$$, D is a moving point inside $$\triangle A B C$$, $$\bigodot O$$ is the circumcircle of $$\triangle A C D$$, intersecting $$B C$$ at point E, line $$B D$$ intersects $$\bigodot O$$ at point P, $$\overset{⌢}{A E} = \overset{⌢}{C P}$$, then the minimum value of $$A D$$ is      .

Answer: $$\sqrt{41} - 4$$


As shown in the figure, quadrilateral $$A B C D$$ is a rectangle, point $$F$$ is a trisection point of side $$A B$$, with $$B F = 2 A F$$, point $$E_{1}$$ is the midpoint of side $$C B$$, connecting $$E_{1} F$$ and $$E_{1} D$$ to form $$\triangle E_{1} F D$$; point $$E_{2}$$ is the midpoint of $$C E_{1}$$, connecting $$E_{2} F$$ and $$E_{2} D$$ to form $$\triangle E_{2} F D$$; point $$E_{3}$$ is the midpoint of $$C E_{2}$$, connecting $$E_{3} F$$ and $$E_{3} D$$ to form $$\triangle E_{3} F D$$; … continuing in this pattern, if the area of rectangle $$A B C D$$ is 6, then the area of $$\triangle E_{2022} F D$$ is       .

Answer: $$3 - \frac{1}{2^{2022}}$$


The ancient Chinese mathematician Zhao Shuang, while annotating the "Zhou Bi Suan Jing," used four congruent right triangles to form a square (as shown in the figure) and employed it to prove the Pythagorean theorem; this diagram is known as the "Zhao Shuang Chord Diagram." Draw $$EM \parallel NG \parallel AD$$, and if $$GF = 2FM$$, then the value of $$\frac{MN}{FD}$$ is        .

Answer: $$\frac{\sqrt{5}}{2}$$/$$\frac{1}{2} \sqrt{5}$$


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E is a moving point on $$CD$$ (point E does not coincide with points C or D). Connect $$AE$$ intersecting $$BD$$ at F. From F, draw $$FH \bot AE$$ intersecting $$BC$$ at point H. From H, draw $$HG \bot BD$$ at G. The following conclusions are given: ① $$AF = FH$$; ② $$\angle HAE = 45^\circ$$; ③ $$BD = 2FG$$; ④ The perimeter of $$\triangle CEH$$ is 9. Among these, the ones that must be true are   .

Answer: ①②③


As shown in the figure, in the isosceles right triangle $$\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A B = A C$$, the altitude $$A H$$ intersects the median $$B D$$ at point F. From A, draw $$A E \bot B D$$ intersecting BC at point E, and connect $$H D$$, yielding the following five conclusions: ① $$\angle A B D = \angle C A E$$, ② $$\triangle A B F \approx \triangle C A E$$, ③ $$\angle E D C - \angle C B D = 30 \circ$$, ④ $$A E + D E < A H + D H$$, ⑤ $$S_{\triangle A B C} = 6 S_{\triangle C D E}$$. Among these, the correct conclusions are      (fill in the serial numbers).

Answer: ①②④⑤


As shown in the figure, in rectangle $$A B C D$$, $$A B = 1$$, $$B C = \sqrt{3}$$, $$E$$ is a moving point on side $$A D$$, connect $$B E$$, $$A F \bot B E$$ at $$F$$, $$G$$ is the midpoint of $$B C$$, connect $$F G$$, construct an equilateral triangle $$\triangle F G H$$ to the right using $$F G$$ as a side, connect $$C H$$, then the minimum value of $$C H$$ is     .

Answer: $$\frac{\sqrt{7} - 1}{2}$$


As shown in the figure, the side length of square ABCD is 1, and AC and BD are its diagonals. Rotate triangle DCB clockwise around point D by 45° to obtain triangle DGH. HG intersects AB at point E, connect DE intersecting AC at point F, and connect FG. The following conclusions are given: ① Quadrilateral AEGF is a rhombus; ② The area of triangle HED is $1 - \frac{\sqrt{2}}{2}$; ③ ∠AFG = 135°; ④ BC + FG = $\sqrt{3}$. Among these, the correct conclusions are ______. (Fill in the correct sequence numbers)

Answer: ①②③


As shown in the figure, in quadrilateral $$A B C D$$, $$A C \bot A B, B D \bot C D, B D = C D$$, if $$A B = m, A C = n$$, and $$m < n$$, then the length of $$A D$$ is          (expressed in terms of $$m, n$$).

Answer: $$\frac{\sqrt{2} \left(n - m\right)}{2}$$


As shown in the figure, in parallelogram $$A B C D$$, $$A D = 8 \text{cm}$$, $$A B = 4 \text{cm}$$, $$A E$$ bisects $$\angle B A D$$ and intersects side $$B C$$ at point $$E$$, and intersects the extension of $$D C$$ at point $$F$$. The following conclusions: ① $$C E = 4 \text{cm}$$; ② segments $$A F$$ and $$B C$$ bisect each other; ③ $$A C \bot D F$$; ④ $$D E \bot A F$$; among these, the correct conclusions are:      (fill in the serial numbers).

Answer: ①②④


Given: As shown in the figure, ∠ABC = 40°, point P is a moving point on ray BC. Triangle ABP is folded along AP, and the image of point B is point D. When line AD is perpendicular to BC, ∠ABD =     °.

Answer: 65 or 25.


As shown in the figure, in △ABC, points D and E are points on sides AB and AC respectively, and BE = CD, CD ⊥ BE. If ∠A = 30°, BD = 1, CE = 2$$\sqrt{3}$$, then the area of quadrilateral CEDB is     .

Answer: $$\frac{19}{4}$$


As shown in the figure, in the right triangle $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, a square $$A B D E$$ is constructed outwardly on the hypotenuse $$A B$$, and the diagonals of the square intersect at point $$O$$. Connect $$O C$$. Given $$A C = 5$$ and $$O C = 6 \sqrt{2}$$, the length of the other leg $$B C$$ is           .

Answer: $$7$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$. The vertices $$E$$, $$F$$, $$G$$, $$H$$ of rhombus $$E F G H$$ lie on sides $$A D$$, $$A B$$, $$B C$$, $$C D$$ respectively. If $$A F : B G = 3 : 4$$, then the length of $$C G$$ is        .

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


As shown in the figure, in the rhombus $$ABCD$$, $$\angle A = 60^\circ$$, points M and N are any two points on sides $$AD$$ and $$AB$$, respectively. The rhombus $$ABCD$$ is folded along $$MN$$, and point A lands exactly at point E on diagonal $$BD$$. The following conclusions: ① $$\triangle MED \sim \triangle ENB$$; ② If $$\angle DME = 20^\circ$$, then $$\angle ENB = 100^\circ$$; ③ If $$DE : BE = 1 : 2$$, then $$AM : AN = 1 : 2$$; ④ If the side length of the rhombus is 4, M is the midpoint of $$AD$$, and segment $$MC$$ is connected, then $$MC = 2\sqrt{7}$$. The correct conclusions are:       (fill in all correct conclusion numbers)

Answer: ①②④


In isosceles triangle △ABC, AB = AC = 4, ∠BAC = 120°. A line l rotates arbitrarily about vertex A. From point C, drop a perpendicular to line l, with foot at H. The range of the distance between points B and H is     .

Answer: $$2 \sqrt{7} - 2$$≤BH≤$$2 \sqrt{7} + 2$$.


As shown in the figure, rectangle $$A B C D$$ is inscribed in $$\bigodot O$$, $$A B = 40$$, $$B C = 55$$, $$E$$ is a point on arc $$B C$$, connect $$A E$$, $$C E$$, $$C E = 20$$, then $$A E =$$      ; $$F$$ is a moving point on ray $$C E$$, $$\angle B F E > 90 \circ$$, when $$B F = A B$$, $$E F =$$      .

Answer: $$65$$     $$24 - \sqrt{511}$$/$$- \sqrt{511} + 24$$


As shown in the figure, $$A C \bot B C$$, $$D C \bot E C$$, $$A C = B C$$, $$D C = E C$$, if $$A C = 2$$, $$C E = 3$$, then $$A D^{2} + B E^{2} =$$      .

Answer: 26


As shown in the figure, in parallelogram ABCD, ∠BAD = 45°, AB = 10. The paper is folded such that the image point A' of point A falls on side BC; the crease EF intersects AB, AD, and AA' at points E, F, and G, respectively. The paper is then folded again such that the image point C' of point C falls on A'F. Connect GC'. The distance from point G to AD is  , and the minimum value of GC' is     .

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


As shown in the figure, the side length of square $$A B C D$$ is $$3$$, points $$P$$ and $$Q$$ lie on the extensions of $$A B$$ and $$B C$$ respectively, and $$B P = C Q$$. Connect $$A Q$$ and $$D P$$, intersecting at point $$O$$, and intersecting sides $$C D$$ and $$B C$$ at points $$F$$ and $$E$$ respectively. Connect $$A E$$. The following conclusions are given: ① $$A Q \bot D P$$; ② $$O A^{2} = O D \cdot O P$$; ③ $$S_{\triangle A O D} = S_{O E C F}$$; ④ When $$B P = 1$$, $$t a n \angle O A E = \frac{11}{16}$$. Among these, the correct ones are      . (Write all the correct conclusion numbers)

Answer: ①②③


As shown in the figure, on the sides $$AC$$ and $$BC$$ of the equilateral triangle $$ABC$$, take points $$P$$ and $$Q$$ such that $$AP = CQ$$. Let $$AQ$$ and $$BP$$ intersect at point $$O$$. If $$BO = 6$$ and $$PO = 2$$, then the length of $$AP$$ is    , and the length of $$AO$$ is       .

Answer: 4     $$1 + \sqrt{13}$$


As shown in the figure, in square ABCD with side length 1, rotate ray AC around point A clockwise by α degrees (0 < α ≤ 360°) to obtain ray AE. Let M be the symmetric point of D with respect to ray AE. Then the minimum value of the length of segment CM is          .

Answer: $$\sqrt{2}$$﹣1


As shown in the figure, in rectangle $$A B C D$$, rotate $$\triangle A D C$$ counterclockwise about point D by $$90 \circ$$ to obtain $$\triangle F D E$$, such that points B, F, and E lie on the same straight line. Let $$A C$$ intersect $$B E$$ at point G, and connect $$D G$$. Which of the following conclusions is correct:     .
①$$A C \bot B E$$; ②$$\triangle B C G \sim \triangle G A D$$; ③$$D F 2 = F C \cdot D C$$; ④$$C G + \sqrt{2} D G = E G$$.

Answer: ①③④


As shown in the figure, in the square $$ABCD$$ with side length 6, point $$E$$ is on diagonal $$AC$$. Connect $$DE$$, and from point $$E$$ draw a perpendicular to $$DE$$ intersecting the line containing side $$AB$$ at point $$F$$. Connect $$DF$$, intersecting the line containing diagonal $$AC$$ at point $$G$$. If $$AF = \frac{1}{2} AB$$, then the length of segment $$EC =$$     .

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


As shown in the figure, through a point $P$ on the side $AB$ of the equilateral triangle $\triangle ABC$ with side length 4, draw $PE \bot AC$ at $E$. Let $Q$ be a point on the extension of $BC$ such that $PA = CQ$. Connect $PQ$ intersecting side $AC$ at $D$. Then the length of $DE$ is           .

Answer: $$2$$


As shown in the figure, parallelogram paper $$ABCD$$ is folded along crease $$MN$$, with point $$M$$ and point $$N$$ on sides $$AD$$ and $$BC$$ respectively. The images of points $$C$$ and $$D$$ are $$E$$ and $$F$$, respectively, and point $$F$$ lies inside the parallelogram. The extension of $$MF$$ intersects $$BC$$ at point $$G$$. $$EF$$ intersects side $$BC$$ at point $$H$$. Given $$AB = 6$$, $$\angle B = 45^\circ$$, $$CN = 2\sqrt{2}$$, and when point $$H$$ is a trisection point of $$NG$$, the length of $$MD$$ is          .

Answer: $$6 \sqrt{5} - 4 \sqrt{2}$$ or $$3 \sqrt{10} - \sqrt{2}$$


As shown in the figure, point E is a point on side BC of square ABCD. Triangle ABE is folded over AE as the axis of symmetry to obtain triangle AFE. Then, AD is folded onto AF, and the crease intersects the extension of BF at point H. BH intersects AE at point G. Connect DH.
(1) The measure of ∠AHB is      ;
(2) If AB = 2, the maximum distance from point H to AB is      .

Answer: 45°     $$1 + \sqrt{2}$$/$$\sqrt{2} + 1$$


As shown in the figure, the side length of rhombus ABCD is 4, ∠BAD = 60°, point E is a moving point on AD (not coinciding with A or D), point F is a moving point on CD, and AE + CF = 4. The minimum area of △BEF is     .

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


As shown in the figure, point O is a point inside the equilateral triangle $$\triangle ABC$$, with $$OA = 3$$, $$OB = 4$$, $$OC = 5$$. Segment $$BO$$ is rotated counterclockwise by $$60^\circ$$ about point B to obtain segment $$BO^{'}$$. The following conclusions: ① $$\triangle BO^{'}A$$ can be obtained by rotating $$\triangle BOC$$ counterclockwise about point B by $$60^\circ$$; ② the distance between point O and $$O^{'}$$ is 4; ③ $$\angle AOB = 150^\circ$$; ④ $$S_{\text{quadrilateral } AOBO^{'}} = 6 + 3\sqrt{3}$$. Among these, the correct conclusions are            .

Answer: ①②③


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 7$$, $$B C = \sqrt{15}$$, point D lies on side $$A B$$, point E lies on side $$A C$$, $$C E = 1$$, fold quadrilateral $$B C E D$$ along line $$D E$$ to obtain quadrilateral $$G F E D$$, with point B mapped to point G, connect $$A G$$ and $$C F$$, when $$A G$$ is minimized, the value of $$\cos \angle E F C$$ is           .

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


As shown in the figure, in △ABC, ∠ACB = 90°, AB = 4, point O is the midpoint of AB. An isosceles right triangle BCD is constructed outwardly with BC as one leg. Connect OD. When OD reaches its maximum value, the measure of ∠ODB is     .

Answer: 22.5°


As shown in the figure, segment AB = 4, M is the midpoint of AB, and point P moves such that its distance to point M is 1. Connect PB; rotate segment PB 90° counterclockwise about point P to obtain segment PC. Connect AC. What is the maximum length of segment AC?

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


As shown in the figure, in rectangle ABCD, E is a point on side BC. Connect AE, and fold triangle ABE along AE; the corresponding point of B is F. If the extension of segment AF passes through the midpoint of one side of the rectangle, and AB = 2, AD = 4, then the length of BE is ______.

Answer: $$2 \sqrt{2} - 2$$ or $$\frac{\sqrt{17} - 1}{2}$$ or 2


As shown in the figure, square $$A B C D$$, point $$E$$ is a moving point on side $$A D$$, point $$A^{'}$$ is the reflection of point $$A$$ over line $$B E$$, connect $$A A^{'}$$ and extend it to intersect $$B E$$ at point $$G$$ and intersect $$C D$$ at point $$F$$, construct $$D H \bot A F$$, given $$G H = 2$$, when point $$A^{'}$$, the reflection of point $$A$$, lies on diagonal $$B D$$, the value of $$\frac{F D}{A D}$$ is        ; the area of square $$A B C D$$ is        .

Answer: $$\sqrt{2} - 1$$     $$8 + 4 \sqrt{2}$$


As shown in the figure, in the rhombus $$A B C D$$, $$A B = 8$$, $$\angle B A D = 60^\circ$$. Inside it, construct two rhombuses $$A E N H$$ and $$C G M F$$ of the same shape and size, such that points E, F, G, H lie on sides $$A B$$, $$B C$$, $$C D$$, $$D A$$ respectively, and points M, N lie on diagonal $$A C$$.
(1) If $$A E = 3 B E$$, then the length of $$M N$$ is           .
(2) If $$A E = B E$$, let P and Q be two moving points on $$D E$$ and $$A D$$ respectively, then the minimum value of $$A P + P Q$$ is           .

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


As shown in the figure, in $$\triangle A B C$$, $$E$$ is the midpoint of $$A B$$, $$D$$ is a point on $$A C$$, connect $$D E$$, $$B H \bot A C$$ at $$H$$, if $$2 \angle A D E = 90 \circ - \angle H B C$$, $$A D : B C = 4 : 3$$, $$C D = 2$$, then the length of $$B C$$ is      .

Answer: 6


In △ABC, ∠A = 45°, ∠B = 60°, AB = 4, points P, M, N are on sides AB, BC, CA respectively, connect PM, MN, NP, then the minimum perimeter of △PMN is            .

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


As shown in the figure, quadrilateral $$A B C D$$ is a square, point $$E$$ lies on the extension of $$C B$$, connect $$A E$$, $$A F \bot A E$$ intersects $$C D$$ at point $$F$$, connect $$E F$$, point $$H$$ is the midpoint of $$E F$$, connect $$B H$$, then among the following conclusions: ① $$B E = D F$$; ② $$\angle B E H = \angle B A H$$; ③ $$\frac{B H}{C F} = \frac{\sqrt{3}}{2}$$; ④ if $$A B = 4$$, $$D F = 1$$, then the area of $$\triangle B E H$$ is $$\frac{3}{4}$$. The correct ones are      . (Fill in all correct conclusion numbers on the line)

Answer: $$\textcircled{1}\textcircled{2}\textcircled{4}$$


As shown in the figure, $$\bigodot O$$ is the circumcircle of $$\triangle A B C$$. Draw $$O D \bot A B$$ through the center $$O$$, intersecting $$A B$$ at point $$D$$. Extend $$D O$$ to intersect $$\bigodot O$$ at point $$F$$. Connect $$C F$$ and $$B F$$. Point $$C$$ is the midpoint of the minor arc $$B F$$. Take a point $$E$$ on the extension of $$A B$$, and connect $$C E$$, such that $$\angle B C E = \angle C A B$$. If $$B C = 10$$ and $$\text{tan} \angle C A B = \frac{3}{4}$$, then the radius of $$\bigodot O$$ is  , and $$\frac{B E}{C E} =$$           .

Answer: $$\frac{25}{3}$$     $$\frac{25}{39}$$


As shown in the figure, in △ABC, AB = AC = 4, BC = 2. Points P, E, F are arbitrary points on sides BC, AB, and AC, respectively. The minimum value of PE + PF is     .

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


In parallelogram $$A B C D$$, $$A C$$ is a diagonal, point $$E$$ is a point on $$A C$$, a line $$G F$$ passing through point $$E$$ intersects side $$C D$$ at point $$F$$ and side $$A B$$ at point $$G$$, connect $$D E$$. If $$D E \bot G F$$, $$D F = 2 E F$$, $$\angle A C B = 60 \circ$$, $$\frac{A C}{B C} = \frac{5}{4}$$, then the value of $$\frac{C F}{D F}$$ is             .

Answer: $$\frac{1}{8}$$/$$0 . 125$$


Given that the side length of square $$A B C D$$ is 4, and point $$P$$ is a point on ray $$B C$$, folding $$\triangle A B P$$ along $$A P$$ yields $$\triangle A E P$$. If $$\triangle C D E$$ is an isosceles triangle, then the distance from point $$E$$ to $$B C$$ could be          .

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


As shown in the figure, in rhombus $$A B C D$$, $$\angle B = 60 \circ$$, point $$E$$ is on side $$A B$$, with $$A E = 4$$, $$B E = 8$$, point $$F$$ is a point on $$B C$$, and $$\triangle E G F$$ is a right triangle with right angle at point $$G$$ and $$\angle E F G = 30^\circ$$. Connect $$A G$$. When point $$F$$ moves along ray $$B C$$,
(1) When $$B F = \frac{1}{2} B E$$, $$A G =$$      .
(2) The minimum value of segment $$D G$$ is      .

Answer: $$2 \sqrt{7}$$     $$4 \sqrt{3}$$


As shown in the figure, the square paper $$ABCD$$ has a side length of 4. Point E is on side $$AD$$, and point F is on side $$CD$$. The square paper $$ABCD$$ is folded along EF, and the image of point B is point G. Connect $$DG$$. If $$AE = 1$$, then the minimum length of $$DG$$ is      .

Answer: $$\sqrt{17} - 3$$


As shown in the figure, the equilateral triangle $$\triangle A B C$$ has side length 2. E and F are two moving points on sides $$B C$$ and $$C A$$, respectively, such that $$B E = C F$$. Connect $$A E$$ and $$B F$$, and let their intersection be P. Then the minimum value of $$P C$$ is          .

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


As shown in the figure, in rectangle $$A B C D$$, $$A C$$ is the diagonal, point F lies on $$A D$$, and segment $$B F$$ intersects $$A C$$ at point E, with $$\angle A B F = \angle F A C$$, and $$\frac{A B}{B C} = \frac{1}{2}$$;

(1) then $$\frac{A F}{F D} =$$          ;
(2) If $$E G = \sqrt{2}$$, $$\triangle A B G$$ is an isosceles right triangle with $$A G = B G$$, then $$G H =$$           .

Answer: $$\frac{1}{3}$$     $$\frac{5 \sqrt{2}}{3}$$


As shown in the figure, point C lies on $$\bigodot O$$, diameter $$AB = 25$$, $$CD \bot AB$$, with foot at D, point E is the incenter of $$\triangle BCD$$, $$AC = 15$$, point F lies on it, $$AF = 2FC$$, then $$EF =$$     .

Answer: $$\sqrt{65}$$


As shown in the figure, draw ray CP on the right side of equilateral triangle ABC, $$\angle A C P = \alpha < 60^\circ$$, point D is the reflection of point A over ray CP, BD intersects CP at point E, connect AD and AE. If $$A E = 3$$, $$C E = 4$$, then $$\angle B E C =$$      , $$B D =$$         .

Answer: 60°     10


As shown in the figure, the four corners of rectangle paper ABCD are folded inward, with EH, EF, FG, GH as the creases, where points A and B coincide at point J, and points C and D coincide at point K, and points H, J, K, F lie on the same straight line.
(1) The shape of quadrilateral EFGH is            .
(2) If $$\frac{A H}{D H} = \frac{3}{4}$$, JK = $$\sqrt{2}$$, then AB =          .

Answer: rectangle;     $$4 \sqrt{6}$$.


As shown in the figure, in parallelogram $$\square ABCD$$, $$AB = BC = 2$$, $$\angle B = 120^\circ$$, point $$E$$ is the reflection of point $$A$$ over line $$MN$$, and points $$N$$, $$E$$, $$M$$ lie on sides $$AB$$, $$BC$$, $$AD$$ respectively. Then the maximum value of $$MD$$ is      .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C$$, point D is any point on side BC, connect AD, from point C draw $$C E \bot A D$$ at point E, from point C draw $$C F \bot C E$$, and $$C F = C E$$, connect FE and extend it to intersect AB at point M, connect BF. If the area of quadrilateral AMEC is 8, $$C E = 2$$, then the area of quadrilateral ABFC is        .

Answer: 18


As shown in the figure, quadrilateral $$A B C D$$ is a rhombus, with diagonals $$A C$$, $$B D$$ intersecting at point O, $$A C = 2 \sqrt{3}$$, $$B D = 2$$, point P is a moving point on $$A C$$, and point E is the midpoint of $$A B$$. Then the minimum value of $$P D + P E$$ is        .

Answer: $$\sqrt{3}$$


As shown in the figure, place a set of triangle rulers as shown, with point E on side AC. Rotate triangle $$\triangle ABC$$ around point A at a speed of 3° per second in a clockwise direction for one full rotation. During the rotation, at the     second, side BC is exactly parallel to side DE.

Answer: 35 or 95


As shown in the figure, in $$\triangle A B C$$, point $$D$$ lies on $$B C$$, connect $$A D$$, $$\angle D A C = 45 \circ$$, point $$E$$ lies on $$A D$$, connect $$C E$$, $$\angle B C E = 45 \circ$$, if $$C E = B C$$, and the area of $$\triangle A B C$$ is $$8 \sqrt{2}$$, then the length of $$A C$$ is    .

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


As shown in the figure, in $$R t \triangle A B C$$, $$\angle C = 90 \circ$$, $$A C = 3$$, $$B C = 4$$, points $$D$$, $$E$$ are the midpoints of sides $$C A$$, $$C B$$ respectively, the angle bisector of $$\angle C A B$$ intersects $$D E$$ at point $$F$$, then the length of $$C F$$ is          .

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


As shown in the figure, in the triangular paper $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A B = 10$$, $$B C > A C$$, points D and E lie on AB. Fold $$\angle A$$ along CD so that point A lands at point $$A^{'}$$ on AB; fold $$\angle B$$ along CE so that point B lands on the extension of $$C A^{'}$$, with B's corresponding point being $$B^{'}$$. If the area of $$\triangle A^{'} E B^{'}$$ is $$\frac{9}{4}$$, then the value of $$A C \cdot B C$$ is        .

Answer: 5 or 45


As shown in the figure, in rectangle ABMN, AN = $$\sqrt{2}$$, point C is the midpoint of MN, connect AC and BC, and BC = 2$$\sqrt{2}$$, point D is the midpoint of AC, point E is a moving point on side AB, connect DE, point F is the reflection of point A over line DE, connect DF and EF. When EF ⊥ AC, the length of AE is    .

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


As shown in the figure, in parallelogram $$A B C D$$, $$A B = 4$$, $$B C = 6$$, $$\angle A B C = 60 \circ$$, point P is a point in the plane satisfying $$\angle A P D = 90 \circ$$, where point Q is a point on segment $$C P$$ such that $$\frac{P Q}{C Q} = \frac{1}{2}$$, connect $$B Q$$, then the minimum value of $$B Q$$ is         .

Answer: $$\frac{4 \sqrt{19}}{3} - 2$$


As shown in the figure, students from the school labor club used mathematical knowledge to sketch the emblem of the club. The design process is as follows: Construct an isosceles triangle $$\triangle ABC$$ inscribed in circle $$O$$, draw $$AD \bot BC$$ intersecting $$BC$$ at point $$E$$ and intersecting circle $$O$$ at point $$D$$, draw $$EF \bot AB$$ at point $$F$$ and $$EG \bot AC$$ at point $$G$$, connect $$DF$$ and $$DG$$. It is measured that $$DE = 2 \text{cm}$$, $$BC = 8 \text{cm}$$. The area of quadrilateral $$AFDG$$ is      $$\text{cm}^{\text{2}}$$.

Answer: $$32$$


As shown in the figure, in rhombus $$ABCD$$ with side length 4, $$\angle BAD = 60^\circ$$. There is a triangle board $$\triangle BEF$$ with $$\angle BFE = 30^\circ$$. When $$\triangle BEF$$ is rotated about point $$B$$ to form $$\triangle BE'F'$$, the lines containing $$BE'$$ and $$BF'$$ intersect segment $$AC$$ at points $$M$$ and $$N$$, respectively. If $$C'$$ is the reflection of point $$C$$ across line $$BE'$$, and when $$C'N \perp AC$$, the length of $$AN$$ is           .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 52^\circ$$, $$A B = A C$$, $$B D \bot A C$$ at D, E and F move along segments $$B D$$ and $$B C$$ respectively, and $$B E = C F$$. When $$A E + A F$$ is minimized, the measure of $$\angle A E D$$ is          .

Answer: $$77 \circ$$


As shown in the figure, in square $$ABCD$$, moving points $$E$$ and $$F$$ start simultaneously from points $$D$$ and $$C$$, respectively, moving at the same speed along sides $$DC$$ and $$CB$$, respectively. Connect $$AE$$ and $$DF$$, intersecting at point $$P$$. As points $$E$$ and $$F$$ move, point $$P$$ also moves accordingly. If $$AD = 2$$, the minimum value of segment $$CP$$ is          .

Answer: $$\sqrt{\text{5}} −\text{1}$$


As shown in the figure, in the rhombus $$A B C D$$, $$A C$$ and $$B D$$ intersect at point $$O$$, the perpendicular bisector of side $$A B$$ intersects $$B D$$ at point $$E$$ and intersects $$A B$$ at point $$F$$, point $$G$$ is the midpoint of side $$C D$$, connect $$E G$$, if $$A C = 8$$, $$B D = 16$$, then the length of segment $$E G$$ is      .

Answer: $$\sqrt{53}$$


As shown in the figure, in the acute triangle $$\triangle A B C$$, $$\angle B A C = 45 \circ$$, and $$A D \bot B C$$ at point D. If $$B D = a$$, $$A D = b$$, then the length of $$C D$$ is      . (Expressed as an algebraic expression in terms of a and b)

Answer: $$\frac{b^{2} - a b}{a + b}$$


As shown in the figure, in $$\Delta A B C$$, $$\angle B = 90 \circ$$, construct the angle bisectors of the interior angle $$\angle A C B$$ and the exterior angle $$\angle D A C$$, and let the lines containing these two angle bisectors intersect at point $$E$$, then $$\angle E =$$    degrees; construct the angle bisectors of $$\angle E A B$$ and $$\angle E C B$$, and let these two angle bisectors intersect at point $$F$$, then $$\angle A F C =$$      degrees.

Answer: 45     67.5


As shown in the figure, ∠AOB = 50°, ∠BOC = 30°, OM = 11, ON = 6. Points P and Q are moving points on OA and OB respectively. The minimum value of MQ + PQ + NP is    .

Answer: $$\sqrt{223}$$


As shown in the figure, it is given that quadrilateral $$A B C D$$ is a square with all four angles right angles and all four sides equal. Point $$E$$ lies on $$B C$$ such that $$C E = \frac{1}{4} B C$$, and point $$F$$ is the midpoint of $$C D$$. Extend $$A F$$ to intersect the extension of $$B C$$ at point $$M$$. The following conclusions: ① $$A B = C M$$; ② $$A E = A B + C E$$; ③ $$S_{\triangle A E F} = \frac{1}{4} S_{A B C F}$$; ④ $$\angle A F E = 90 \circ$$, among which the correct conclusions are      .

Answer: ①②④


As shown in the figure, in $$\triangle A B C$$, $$A C = 3$$, $$B C = 4$$, $$A B = 5$$. Quadrilateral $$A B E F$$ is a square. Point D is a point on line $$B C$$ such that $$C D = 1$$. Point P is a point on segment $$D E$$ such that $$P D = \frac{2}{3} D E$$. Draw line a through point P parallel to $$B C$$, intersecting segment $$A B$$ at point G and segment $$A D$$ at point H. Then the length of $$G H$$ is           .

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


As shown in the figure, in square $$A B C D$$, point $$E$$ is a point on $$C D$$, and point $$F$$ is a point on the extension of $$C B$$, with $$D E = B F$$. Connect $$B D$$ and $$E F$$, intersecting at point $$G$$, and connect $$A G$$. Among the following conclusions, ① $$\triangle A E F$$ is an isosceles right triangle; ② $$A G \bot E F$$; ③ $$C E = 2 B G$$; ④ $$A B - B F = \sqrt{2} B G$$; ⑤ when $$E C = 3 D E$$, $$\text{sin} \angle D G E = \frac{1}{4}$$. The correct ones are          .

Answer: ①②④


As shown in the figure, rays $$AM$$, $$BN$$ are both perpendicular to segment $$AB$$. From point A, draw perpendicular $$AC$$ to $$BE$$, intersecting $$BE$$ and $$BN$$ at points F and C respectively, with foot at D. If $$CD = CF$$, then $$\frac{AE}{AD} =$$                    .

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


As shown in the figure, $$O$$ is the intersection point of the diagonals $$AC$$ and $$BD$$ of square $$ABCD$$, and $$E$$, $$F$$ are points on sides $$AB$$ and $$BC$$ respectively, such that $$BE + EF = FC$$; then $$\angle EOF =$$      degrees; if $$OE = \frac{\sqrt{6}}{2} OF$$, then $$\frac{AE}{CF} =$$      .

Answer: 45     $$\frac{3}{2}$$


As shown in the figure, given that the side length of square $$A B C D$$ is 2, point $$E$$ is the midpoint of side $$C D$$, and $$\triangle A D E$$ is folded along $$A E$$ to $$\triangle A F E$$, extending $$A F$$ intersects side $$B C$$ at point $$G$$, then $$C G =$$      ; if extending $$E F$$ intersects side $$B C$$ at point $$Q$$, then $$t a n \angle E Q C =$$      .

Answer: $$\frac{1}{2}$$/$$0.5$$     $$\frac{3}{4}$$/$$0.75$$


As shown in the figure, $$C$$, $$D$$ are the trisection points of $$AB$$, arcs are drawn with centers at $$C$$ and $$D$$ and radius equal to $$CD$$, intersecting at points $$E$$ and $$F$$, and $$EF$$ is connected. If $$AB = 9$$, then the length of $$EF$$ is        .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 3$$, point $$E$$ is any point on side $$C D$$ (point $$E$$ does not coincide with points $$C$$ and $$D$$). From point $$A$$, draw $$A F \bot A E$$, intersecting the extension of side $$C B$$ at point $$F$$. Connect $$E F$$, intersecting side $$A B$$ at point $$G$$. Let $$D E = x$$, $$B F = y$$. Then the function expression of $$y$$ in terms of $$x$$ is      ; when point $$E$$ moves along side $$C D$$, if $$\triangle A E G$$ is an isosceles triangle, the length of segment $$D E$$ is      .

Answer: $$y = \frac{4}{3} x \left(0 < x < 4\right)$$     $$\frac{9}{4}$$ or $$\frac{3}{2}$$ or $$\frac{7}{8}$$


As shown in the figure, in right triangle ABC, ∠ACB = 90°, AC = 5 cm, BC = 12 cm. Rotate triangle ABC clockwise around point B by 60° to obtain triangle BDE. Connect DC and let it intersect AB at point F. The sum of the perimeters of triangles ACF and BDF is ______ cm.

Answer: 42


As shown in the figure, in the rhombus $$A B C D$$, $$A B = 4$$, a circle $$\bigodot O$$ is drawn with $$A B$$ as its diameter, intersecting $$B C$$ at point E. A tangent line $$A F$$ to $$\bigodot O$$ is drawn through point A, intersecting $$C D$$ at point F. Connect $$B F$$ and $$A E$$. If $$\angle E A F = 60^\circ$$, then the length of $$B F$$ is       .

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


As shown in the figure, in trapezoid $$A B C D$$, $$A D \parallel B C$$, $$\angle A = 90 \circ$$, $$A D = 3$$, $$A B = 4$$, $$B C = 6$$. Point E lies on side $$B C$$, and triangle $$A B E$$ is folded along $$A E$$, with point B mapping to point F. If $$E F \bot C D$$, then the length of $$E C$$ is      .

Answer: $$\frac{14}{3}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = 20$$, $$A C = 16$$, $$B C = 12$$, a semicircle is drawn with the midpoint O of side AB as its center, tangent to AC. Points P and Q are moving points on side BC and the semicircle, respectively. Connect PQ; the minimum length of PQ is      .

Answer: 2


As shown in the figure, in $$\triangle A B C$$, $$\angle A B C = 90 \circ$$, $$B D$$ is the median of $$\triangle A B C$$, $$D E \bot B C$$ at point E. If $$B E = 3$$, $$D E = 4$$, then the area of quadrilateral $$A B E D$$ is         .

Answer: 18


As shown in the figure, in $$\triangle A C D$$ and $$\triangle B D C$$, $$\angle A = \angle B = 90 \circ$$, $$A D = B C$$, $$\angle A C D = 35 \circ$$, then $$\angle A C B =$$        $$\circ$$.

Answer: $$20$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$B C = 16$$. Fold rectangle $$A B C D$$ along $$E F$$ so that point C coincides with point A. Then the length of the crease $$E F$$ is     .

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


As shown in the figure, in right triangle ABC, ∠C = 90°, ∠BAC = 60°, point D is a point on side BC. Connect AD, fold triangle ACD along AD so that point C lands at point E. When triangle BDE is a right triangle, the measure of ∠CAD is           .

Answer: $$30 \circ$$ or $$45 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$D E$$ is the perpendicular bisector of $$B C$$, intersecting $$A B$$ at point D and $$B C$$ at point E. If $$A B = 8$$, $$A C = 6$$, then the perimeter of $$\triangle A C D$$ is      .

Answer: 14


As shown in the figure, in quadrilateral $$A B C D$$, connect $$A C$$, $$\angle D A B$$ and $$\angle B$$ are complementary, $$P M$$ and $$P N$$ are the midlines of $$\triangle A D C$$ and $$\triangle C B A$$ respectively, connect $$M N$$, $$P M = P N$$, if $$A B = 5$$, $$C D = 2$$, then the minimum value of segment $$A D$$ is      .

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


As shown in the figure, quadrilateral $$A B C D$$ is a rectangle. Connect $$A C$$. Points $$E$$ and $$F$$ are the midpoints of sides $$A B$$ and $$A D$$, respectively. Connect $$E F$$, $$E F = \sqrt{5}$$. $$A M \bot A C$$ intersects the extension of $$C B$$ at point $$M$$. Point $$N$$ is the midpoint of $$C M$$. Connect $$A N$$. If $$t a n \angle A M C = 2$$, then $$A N =$$

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


As shown in the figure, quadrilateral ABCD is a rhombus with AB = 8 and ∠ABC = 60°. M is any point on diagonal BD (excluding point B). The minimum value of AM + $$\frac{1}{2}$$BM is     .

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


As shown in the figure, the cross-section of a certain dam is a trapezoid $$ABCD$$, with $$AE$$ and $$DF$$ as the heights of the trapezoid. The slope angle of the upstream slope $$AB$$ is $$\alpha = 45^\circ$$, and the slope length $$AB = 6\sqrt{2}$$ m. The slope ratio of the downstream slope $$CD$$ is $$i = 1 : \sqrt{3}$$. Then the length of the downstream slope is      m.

Answer: 12


As shown in the figure, in $$\triangle A B C$$, $$A C = B C$$. With point C as the center and an appropriate length as the radius, draw arcs intersecting $$A C$$ and $$B C$$ at points E and F, respectively. Then, with points E and F as centers and a radius greater than $$\frac{1}{2} E F$$, draw arcs intersecting at point G. Draw ray $$C G$$ intersecting $$A B$$ at point D. Through point D, draw $$D H \parallel B C$$ intersecting $$A C$$ at point H. If $$C H = a$$, then $$B C =$$      (expressed as an algebraic expression in terms of a).

Answer: $$2 a$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, chord $$MN \parallel AB$$, perpendiculars are drawn from $$M$$ and $$N$$ to $$AB$$, with feet at $$C$$ and $$D$$, respectively. The following conclusions

① $$AC = BD$$;
② $$\overset{⌢}{AM} = \overset{⌢}{BN}$$;
③ If quadrilateral $$MC DN$$ is a square, then $$MN = \frac{1}{2} AB$$;
④ If $$M$$ is the midpoint of arc $$AN$$, then $$D$$ is the midpoint of $$OB$$.
The sequence numbers of all correct conclusions are      .

Answer: ①②④


As shown in the figure, in $$\triangle A B C$$, $$\angle C A B = 60 \circ$$. With $$A$$ as the center, draw an arc of appropriate length as the radius, intersecting $$A B$$ and $$A C$$ at points $$D$$ and $$E$$, respectively. Then, with $$D$$ and $$E$$ as centers, draw arcs with radii greater than $$\frac{1}{2} D E$$, intersecting at point $$F$$. Draw ray $$A F$$, intersecting $$B C$$ at point $$P$$. Through point $$P$$, draw lines parallel to $$A C$$ and $$A B$$, intersecting $$A B$$ and $$A C$$ at points $$M$$ and $$N$$, respectively. If $$A P = 4 \sqrt{3}$$, then the area of quadrilateral $$A M P N$$ is      .

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


As shown in the figure, in $$\bigodot O$$, point $$C$$ is the midpoint of $$\overset{⌢}{A B}$$, and connecting $$O C$$ intersects chord $$A B$$ at point $$D$$. If $$O D = 3$$ and $$D C = 2$$, then the length of $$A B$$ is      .

Answer: 8.


As shown in the figure, rectangle $$A B C D$$, point E is a point on $$A B$$, connect $$C E$$, take a point F on $$C E$$, connect $$B F$$, draw a perpendicular from F to $$C E$$ intersecting $$A D$$ at point H. If $$\angle E B F + \angle B C E = 90 \circ$$, $$C E = 2 F H$$, $$A D = 6$$, $$C E = 2 \sqrt{10}$$, then the length of $$C D$$ is           .

Answer: 4


As shown in the figure, in sector $$A O B$$, $$\angle A O B = 120 \circ$$, $$A O = 6$$, point $$C$$ is the midpoint of arc $$A B$$, connect $$O C$$, intersecting $$A B$$ at point $$D$$, $$E$$ is the midpoint of $$O D$$, connect $$B E$$, then the area of the shaded region in the figure is       . (Express the result in terms of radicals and $$\pi$$)

Answer: $$6 \pi - \frac{9}{4} \sqrt{3}$$


The ancient Greek mathematician Euclid, in his deep study of proportion theory, proposed the "extreme and mean ratio" problem of dividing a line segment: point G divides a line segment into two parts such that the longer segment is the geometric mean of the entire length and the shorter segment, i.e., it satisfies $$\frac{MG}{MN} = \frac{GN}{MG} = \frac{\sqrt{5} - 1}{2}$$. Later generations called the number $$\frac{\sqrt{5} - 1}{2}$$ the "golden ratio" and referred to point G as the "golden ratio" point of the segment. As shown in the figure, in △ABC, given AB = AC = 3, BC = 4, if D and E are two "golden ratio" points on the sides, then the area of △ADE is      .

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


As shown in the figure, in the equilateral triangle △ABC, take a point P such that the lengths of PA, PB, and PC are m, m, and $$\sqrt{2} m$$, respectively. Rotate segment BP clockwise 60° about point B to obtain segment BQ, and connect CQ. Then the measure of $$\angle A P B$$ is      .

Answer: $$150 \circ$$


In rectangle $$A B C D$$, $$A B = 6$$, $$A D = 8$$, $$P$$ is a moving point on side $$C D$$, connect $$A P$$, draw $$P M \bot A P$$, on $$A P$$ take $$P M = \frac{3}{4} P N$$, draw $$P H \bot M N$$ at $$H$$, connect $$D H$$, then the minimum value of $$D H$$ is       .

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


The data for the three sides of the triangle are shown in the figure; then the range of $$x$$ is           .

Answer: $$- 2 < x < 0$$


As shown in the figure, the diagonals $AC$ and $BD$ of square $ABCD$ intersect at point $O$. Point $E$ is a point on $AD$. Connect $EO$ and extend it to intersect $BC$ at point $F$. If $\angle DOE = 22.5^{\circ}$ and $BF = 2 - \sqrt{2}$, then the perimeter of square $ABCD$ is        .

Answer: $$8$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$, point $$P$$ is a moving point inside rectangle $$A B C D$$, and $$S_{\Delta P A B} = S_{\Delta P C D}$$, then the minimum value of $$P C + P D$$ is     .

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


As shown in the figure, the diagonals of rhombus $$A B C D$$ intersect at point $$O$$. If $$A C = 24$$, $$B D = 10$$, then the perimeter of rhombus $$A B C D$$ is          .

Answer: $$52$$


As shown in the figure, the rhombus $$A B C D$$ has side length 4, $$\angle A = 60 \circ$$, and when folded along $$E F$$, vertex C lands exactly at the midpoint G of side $$A B$$. Then $$B F =$$        .

Answer: $$1.2$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 8$$, $$B C = 6$$, point $$D$$ is a moving point on side $$A B$$, and $$\triangle A C D$$ is folded along side $$C D$$ to obtain $$\triangle C D E$$. When the overlapping part of $$\triangle C D E$$ and $$\triangle A B C$$ is a right triangle, the length of $$A D$$ is       .

Answer: 4 or $$\frac{32}{5}$$


As shown in the figure, in $$\Delta A B C$$, $$s i n B = \frac{1}{4}$$, $$t a n C = \frac{1}{2}$$, $$A B = 4$$, then the length of $$A C$$ is        .

Answer: $$\sqrt{5}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3 \text{cm}$$, $$B C = 4 \text{cm}$$. The rectangle is folded so that point $$C$$ coincides with point $$A$$. The length of the crease is       .

Answer: $$\frac{15}{4}$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$. $$BD$$ bisects $$\angle ABC$$, $$DB$$ intersects $$AC$$ at point $$E$$, $$\angle DBA = 30^\circ$$. If $$BC = 2$$, then the area of $$\triangle BCE$$ is      .

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


As shown in the figure, the diagonals $$AC$$ and $$BD$$ of rhombus $$ABCD$$ intersect at point O, $$BE \parallel AC$$, $$AE \parallel BD$$, and $$OE$$ intersects $$AB$$ at point F. If $$OE = 5$$ and $$AC = 8$$, then the area of rhombus $$ABCD$$ is        .

Answer: 24


As shown in the figure, in the rhombus $$A B C D$$, $$A B = 5$$, $$A C = 8$$, point $$P$$ is a moving point on side $$B C$$, and triangle $$\triangle A B P$$ is folded along $$A P$$ such that vertex $$B$$ lands at point $$B^{'}$$ inside the rhombus $$A B C D$$. When points $$P$$, $$B^{'}$$, and $$D$$ are collinear, the distance from point $$A$$ to line $$P D$$ is      .

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


Given that OA is the radius of ⊙O, extend AO to point B such that OB = 3OA = 3. Construct an isosceles right triangle BMC with right angle at B, such that point M always lies on ⊙O (as shown in the figure). Connect OC; then the maximum value of OC is      .

Answer: $$3 \sqrt{2} + 1$$/$$1 + 3 \sqrt{2}$$


As shown in the figure, in parallelogram ABCD, AB = 3, AD = 4, ∠ABC = 60°. Through the midpoint E of BC, draw EF ⊥ AB, with foot at point F, intersecting the extension of DC at point H. The area of triangle DEF is     .

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


As shown in the figure, in parallelogram ABCD, a line passing through the midpoint O of diagonal AC intersects BC and AD at points E and F, respectively. Only one additional condition is needed to prove that quadrilateral AECF is a rectangle; this condition can be        (write one such condition).

Answer: AC = FE or AE ⟂ BC, etc. (The answer is not unique; any condition that satisfies the problem statement is acceptable.)


As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ bisects $$\angle B A D$$, $$B C = C D = 2$$, $$A B = 5$$, $$A D = 3$$, then the length of $$A C$$ is       .

Answer: $$\sqrt{19}$$


In rectangle $$A B C D$$, $$A B = 3$$, $$A D = 3 \sqrt{3}$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, from point $$A$$ draw $$A E \bot B O$$, with foot at $$E$$, $$N$$ is the midpoint of $$A D$$, connect $$B N$$ intersecting $$A E$$ at point $$P$$, then the length of $$P E$$ is        .

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


As shown in the figure, in the rhombus $$A B C D$$, $$\angle A B C = 120 \circ$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$B D = 8$$, point $$E$$ is the midpoint of $$O D$$, point $$F$$ is a point on $$A B$$ such that $$A F = 3 B F$$, point $$P$$ is a moving point on $$A C$$, connect $$P E$$, $$P F$$, then the maximum value of $$\left|P F - P E\right|$$ is        .

Answer: $$2$$