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

Answer: $$8$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A D$$ is the altitude, $$A B > A C$$, E and F are the midpoints of $$A B$$ and $$B C$$ respectively. If $$\angle C = \alpha$$, then the measure of $$\angle D E F$$ is      (expressed in terms of $$\alpha$$).

Answer: $$2 \alpha - 90 \circ$$


As shown in the figure, the side length of square $$A B C D$$ is 3, and points E, F, G lie on sides $$A B, B C, C D$$ respectively, with $$A F \bot E G$$. When $$C F = 2 B F$$, the minimum value of $$E F + A G$$ is     .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 8$$, $$t a n \angle B A C = \frac{3}{2}$$, point $$D$$ is the midpoint of side $$B C$$, point $$E$$ is any point on side $$A B$$ (point $$E$$ does not coincide with point $$B$$). Fold $$\triangle D B E$$ along $$D E$$ so that point $$B$$ lands at point $$F$$, and connect $$A F$$. When the length of segment $$A F$$ is minimized, the distance between points $$B$$ and $$F$$ is      .

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


As shown in the figure, M is the midpoint of side BC in the obtuse triangle ABC. The line MN passing through M divides triangle ABC into two parts with equal perimeters. Given AB = 8 and angle A = 120°, find MN.

Answer: 4


As shown in the figure, in rectangle $$ABCD$$, $$AB = 4 \text{ cm}$$, $$BC = 3 \text{ cm}$$, point E is the midpoint of $$CD$$, and point P moves from point A at a speed of 1 cm per second along $$A \rightarrow B \rightarrow C \rightarrow E$$, finally reaching point E. If the time taken by point P is $$x$$ seconds, then when $$x =$$        seconds, the area of $$\triangle APE$$ equals $$5 \left(\text{cm}\right)^2$$.

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


As shown in the figure, after folding rectangle paper $$ABCD$$ along $$EF$$, point D and point C land at points $$D'$$ and $$C'$$ respectively, and the extension of $$ED'$$ passes exactly through point B. If $$DE = DC = 3$$ and $$CF = 2$$, then $$AE$$ equals           .

Answer: 4


As shown in the figure, it is given that $$\angle A O B = 60 \circ$$, with a circle centered at point $$O$$ intersecting the two sides of the angle at points C and D. With $$O$$ as the center and a radius greater than $$\frac{1}{2} C D$$, two arcs intersect at a point $$P$$ inside $$\angle A O B$$. Connect $$O P$$. Draw line $$P E \parallel O A$$ intersecting $$O B$$ at point $$E$$, and draw line $$P F \parallel O B$$ intersecting $$O A$$ at point $$F$$. Given $$O P = 6 \text{cm}$$, the area of quadrilateral $$P F O E$$ is      .

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


As shown in the figure, quadrilateral $$A B C D$$ is a cyclic quadrilateral inscribed in $$\bigodot O$$, $$D B$$ bisects $$\angle A D C$$, connect $$O C$$, $$B D$$, $$O C \bot B D$$, if $$\angle A$$ equals $$50 \circ$$, then the measure of $$\angle A D B$$ is         .

Answer: $$25 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 120 \circ$$, $$A B = 6$$, $$A C = 4$$, point $$M$$ is a moving point on side $$A B$$, connect $$C M$$, and let $$\bigodot O$$ with diameter $$A M$$ intersect $$C M$$ at point $$N$$. Then the minimum value of segment $$B N$$ is      .

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


As shown in the figure, in parallelogram $$\square A B C D$$, it is given that $$A B = 4$$, $$B C = 6$$, $$\angle A B C = 60^\circ$$. Point P is a moving point on side $$B C$$ (point P does not coincide with B or C). Connect $$A P$$, and construct point Q as the reflection of point B over line $$A P$$. Then the minimum value of segment $$Q C$$ is            .

Answer: $$2 \sqrt{7} - 4$$/$$- 4 + 2 \sqrt{7}$$


As shown in the figure, the radius of $$\bigodot O$$ is 3, $$AB$$ is the diameter. Arcs are drawn with centers at points $$A$$ and $$B$$ and radius equal to the length of $$AB$$. These arcs intersect at points $$C$$ and $$D$$. The area of the shaded region in the figure is      .

Answer: $$15 \pi - 18 \sqrt{3}$$


As shown in the figure, the parallelogram paper $$A B C D$$ is folded along the diagonal $$A C$$, and point B lands at point $$B^{'}$$, with $$B^{'} C$$ intersecting $$A D$$ at point E. At this time, $$\triangle C D E$$ is exactly an equilateral triangle. If $$A B = 6 \text{cm}$$, then $$A D =$$       $$\text{cm}$$.

Answer: 12


As shown in the figure, given isosceles right triangle $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A B = \sqrt{2}$$, point C is a common vertex of rectangle $$B C G F$$ and $$\triangle A B C$$, and $$C E = 1$$, $$C G = 3$$; point D is a point on the extension of $$C B$$ such that $$C D = 2$$. Connect $$B G$$ and $$D F$$. During the process in which rectangle $$E C G F$$ rotates clockwise around point C for one full revolution, when segment $$B G$$ reaches its maximum and minimum lengths, the corresponding lengths of segment $$D F$$ are m and n, respectively. Find the value of $$\frac{m}{n}$$.

Answer: $$\sqrt{13}$$


As shown in the figure, quadrilateral ABCD is a rectangle, $$AB = 3$$, $$BC = 4$$, point P is a moving point on segment $$BC$$, point M is a point on segment $$AP$$, $$\angle ADM = \angle BAP$$, then the minimum value of $$BM$$ is      .

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


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, point $$C$$ is a point on the circle, $$\angle BAC = 20^\circ$$, and the minor arc $$\overset{⌢}{AC}$$ is reflected across the line containing chord $$AC$$, intersecting $$AB$$ at point $$D$$. Then the measure of $$\angle ACD$$ is      .

Answer: $$50 \circ$$


As shown in the figure, in square $$A B C D$$, points E and F lie on sides $$B C$$ and $$C D$$ respectively, $$A E = A F$$, $$\angle E A F = 30^{\circ}$$, then $$\angle A E B =$$       $$\circ$$.

Answer: $$60$$


As shown in the figure, in parallelogram $$\square ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point O. A line passing through point O intersects $$AD$$ and $$BC$$ at points M and N, respectively. If the area of $$\triangle CON$$ is 2 and the area of $$\triangle DOM$$ is 4, then the area of $$\square ABCD$$ is          .

Answer: 24


As shown in the figure, extend the sides of the regular pentagon $$F G H J K$$ such that $$G A = H B = J C = K D = F E$$; if $$A B = A F$$, then the measure of $$\angle B C H$$ is         .

Answer: $$36 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$A C = B C$$, $$\angle A C B = 90 \circ$$. D, E, F are points on sides $$A B$$, $$A C$$, $$B C$$ respectively, and $$C E = C F$$. If $$A D = 2$$, $$B D = 1$$, then the minimum value of $$D E + D F$$ is          .

Answer: $$\sqrt{5}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 2$$, D is the midpoint of $$A B$$, and $$\triangle B C D$$ is folded along the line $$C D$$, with point B landing at point E. If $$C E \bot A B$$, then $$B E =$$         .

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


Point F is the midpoint of side DE of the regular pentagon $$A B C D E$$, connect $$B F$$ and extend it to intersect the extension of $$C D$$ at point G, then the measure of $$\angle B G C$$ is      .

Answer: $$18 \circ$$


As shown in the figure, the side length of square $$A B C D$$ is 4, point $$E$$ lies on side $$B C$$, $$B E = \frac{3}{2}$$, and an isosceles right triangle $$A E F$$ is constructed with $$\angle A E F = 90^\circ$$.

(1) The length of $$C F$$ is      . (2) If $$M$$ is the midpoint of $$A F$$, and segment $$D M$$ is connected, then the length of $$D M$$ is      .

Answer: $$\frac{3 \sqrt{2}}{2}$$     $$\frac{5 \sqrt{2}}{4}$$


As shown in the figure, the rhombus $$A B C D$$ has side length 6, $$\angle B A D = 120 \circ$$, and from point $$D$$, draw $$D E \bot B C$$, intersecting the extension of $$B C$$ at point $$E$$. Connect $$A E$$, intersecting $$B D$$ and $$C D$$ at points $$F$$ and $$G$$, respectively. Then the length of $$F G$$ is      .

Answer: $$\frac{4 \sqrt{7}}{5}$$/$$\frac{4}{5} \sqrt{7}$$


As shown in the figure, $$PA$$ is tangent to $$\bigodot O$$ at point A, $$PO$$ intersects $$\bigodot O$$ at point B, point C lies on $$PA$$, and $$CB = CA$$. If $$OA = 5$$, $$PA = 12$$, then the length of $$CA$$ is      .

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


In $$\triangle A B C$$, $$\angle A B C = 60 \circ$$, $$A B = 9$$, point D is a point on side AB, $$B D = B C$$, connect CD, fold $$\triangle A D C$$ along CD to obtain $$\triangle A_{1} D C$$, where $$A_{1} C$$ intersects side AB at point E, $$B E = 4$$, connect $$A_{1} B$$, then the length of $$A_{1} B$$ is      .

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


As shown in the figure, quadrilateral $$A B C E$$ is inscribed in $$\bigodot O$$, connect $$A C$$, $$A C$$ is the diameter of $$\bigodot O$$, $$E$$ is the midpoint of $$\overset{⌢}{A C B}$$. Draw the tangent $$E F$$ to $$\bigodot O$$ through point $$E$$, intersecting the extension of $$B C$$ at point $$F$$, and $$E F \bot B C$$, $$E F = 4$$, $$B F = 5$$, then the length of $$A E$$ is    , and the radius of $$\bigodot O$$ is       .

Answer: $$\sqrt{41}$$     $$\frac{41}{10}$$


As shown in the figure, hexagon $$A B C D E F$$ is a regular hexagon inscribed in $$\bigodot O$$. Let the area of the regular hexagon $$A B C D E F$$ be $$S_{1}$$, and the area of $$\triangle A C E$$ be $$S_{2}$$. Then $$\frac{S_{1}}{S_{2}} =$$         .

Answer: 2


As shown in the figure, in $$\triangle A B C$$ and $$\triangle A D E$$, $$A B = A C$$, $$\angle B A C = \angle D A E = 40 \circ$$. Rotate $$\triangle A D E$$ clockwise around point A by a certain angle. When $$A D \bot B C$$, the measure of $$\angle B A E$$ is      .

Answer: $$60 \circ$$ or $$120 \circ$$


As shown in the figure, in the square $$ABCD$$ with side length 6, point M is the midpoint of $$AB$$, and point E lies on $$AD$$ such that $$AE = \frac{1}{3} AD$$. In the isosceles triangle $$EDF$$, $$ED = FD$$, and $$\angle EDF = 120^\circ$$.

(1) The area of $$\triangle EDF$$ is      ; (2) If N is the midpoint of $$EF$$, then the value of $$MN^2$$ is      .

Answer: $$4 \sqrt{3}$$     $$37 - 6 \sqrt{3}$$/$$- 6 \sqrt{3} + 37$$


As shown in the figure, in triangle paper $$A B C$$, $$A B = A C, \angle B = 20 \circ$$, point $$D$$ is a moving point on side $$B C$$, fold the triangle paper along $$A D$$ so that point $$B$$ lands at point $$B^{'}$$, when $$B^{'} D \bot B C$$, the degree measure of $$\angle B A D$$ is           .

Answer: $$25 \circ$$ or $$115 \circ$$


As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C = 6$$, $$B D = 8$$, and from point A, draw $$A E \bot C D$$ intersecting at point E, then $$A E$$ is      .

Answer: $$\frac{2 \text{4}}{\text{5}}$$


An exterior angle of a triangle is 100°, then the angle formed by the bisectors of the two non-adjacent interior angles (obtuse angle) is      .

Answer: 130°


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, and point $$C$$ is a point on $$\bigodot O$$. The minor arc $$BC$$ is folded along chord $$BC$$ and intersects $$AB$$ at point $$D$$. Point $$E$$ is the midpoint of arc $$ACB$$. If $$AC = \sqrt{10}$$ and $$AD = 2$$, then the radius of $$\bigodot O$$ is $$r =$$        ; the area of $$\triangle BCE$$ is        .

Answer: 5     15


As shown in the figure, in rectangle ABCD, AB = 6, AD = 4. Points E and F are moving points on AB and DC respectively, with EF ∥ BC. The minimum value of AF + CE is      .

Answer: 10


As shown in the figure, in triangle $$A B C$$, $$\angle A C B = 90 \circ$$, $$A B = 10$$, $$A C = 8$$, points $$D$$ and $$E$$ are points on segments $$A C$$ and $$A B$$ respectively, and connect $$D E$$. Fold $$\triangle A D E$$ along $$D E$$ so that point $$A$$ lands at point $$F$$ on the extension of $$B C$$, at which time $$\angle B F E = 30 \circ$$, then the length of $$C F$$ is      .

Answer: $$\frac{40 \sqrt{3} - 48}{13}$$


As shown in the figure, in rectangle ABCD, point E lies on side BC. Triangle CDE is folded along DE to obtain triangle FDE, with point F landing on AE. If CE = 3 cm and AF = 2 EF, then AB =      cm.

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


As shown in the figure, $$AC$$ bisects $$\angle DCB$$, $$CB = CD$$, the extension of $$DA$$ intersects $$BC$$ at point $$E$$, if $$\angle EAC = 49^{\circ}$$, then the measure of $$\angle BAE$$ is          .

Answer: $$82 \circ .$$


As shown in the figure, on the side $$AD$$ of the square $$ABCD$$, there is a point $$E$$. Connect $$BE$$, and from point $$E$$, draw $$EF \bot BE$$ (point $$F$$ lies to the right of side $$CD$$), with foot at point $$E$$. $$EF$$ intersects $$CD$$ at point $$G$$. Connect $$DF$$. If $$\angle CDF = 45^\circ$$, point $$G$$ is the midpoint of $$EF$$, and $$DG = 1$$.

(Ⅰ) The length of segment $$DE$$ is     ; (Ⅱ) The length of segment $$BE$$ is     .

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


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, E is the midpoint of $$AB$$, connect $$DE$$, rotate $$\triangle DAE$$ counterclockwise about point D by $$90^\circ$$ to obtain $$\triangle DCF$$, connect $$EF$$, then the length of $$EF$$ is      .

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


As shown in the figure, semicircle $$O$$, with point $$O$$ as the center and diameter $$AB$$ of length 6, draw an arc with point $$B$$ as the center and $$OB$$ as the radius, intersecting arc $$AB$$ at point $$C$$. The area of the shaded region is      .

Answer: $$\frac{3}{2} \pi + \frac{9}{4} \sqrt{3}$$


As shown in the figure, in sector $$B O C$$, $$\angle B O C = 60^\circ$$, $$O D$$ bisects $$\angle B O C$$ and intersects arc $$B C$$ at point $$D$$. Point $$E$$ is a moving point on radius $$O B$$. If $$O B = 2$$, then the minimum value of the perimeter of the shaded region is          .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle A = 60 \circ$$, $$A C = 2$$, point $$D$$ is the midpoint of side $$A B$$, and an arc $$B C$$ is drawn with point $$D$$ as the center and $$B D$$ as the radius. The area of the shaded region in the figure is      .

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


As shown in the figure, in $$\triangle A B C$$, $$C A = C B$$, $$\angle A C B = 90 \circ$$, $$A B = 2$$, point $$D$$ is the midpoint of $$A B$$, and a sector $$E D F$$ with center $$D$$ and central angle $$90 \circ$$ is drawn; point $$C$$ lies exactly on $$\overset{⌢}{E F}$$. The area of the shaded region in the figure is      .

Answer: $$\frac{\pi}{4} - \frac{1}{2}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, the circle $$\bigodot O$$ with diameter AB intersects BC at point D. Draw the tangent DE to $$\bigodot O$$ through point D, intersecting the extension of AB at point E. If $$A B = 4$$ and $$\tan \angle C A B = 2$$, then the length of the tangent DE is      .

Answer: $$4$$


As shown in the figure, in square $$ABCD$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, point $$E$$ lies on the extension of $$CB$$, connect $$AE$$, point $$F$$ is the midpoint of $$AE$$, connect $$OF$$ intersecting $$AB$$ at point $$G$$, connect $$BF$$. If $$BE = 3$$, $$OF = \frac{9}{2}$$, then the length of $$BF$$ is      .

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


As shown in the figure, given square $$A B C D$$, point $$E$$ is the midpoint of $$A B$$, connect $$D E$$. $$E F \bot D E$$ intersects $$B C$$ at point $$K$$, and $$E F = D E$$, connect $$D F$$ intersecting $$B C$$ at point $$H$$. $$F G \bot A B$$ intersects the extension of $$A B$$ at point $$G$$. Find the value of $$C H : H K : B K$$ is      .

Answer: $$4 : 5 : 3$$


As shown in the figure, in parallelogram $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point O, and $$O A = O B$$, $$\angle O A D = 65^\circ$$, then $$\angle O D C =$$        .

Answer: $$25 \circ$$


As shown in the figure, a square $ABCD$ with side length $\sqrt{2}$ is inscribed in $\bigodot O$. Tangents to $\bigodot O$ are drawn through points $A$ and $D$, intersecting at point $P$. The area of the shaded region in the figure is        .

Answer: $$1 - \frac{\pi}{4}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ, \angle A = 30 \circ, A B = 8$$. If point D lies on line $$A B$$ (not coinciding with points A or B), and $$\angle B C D = 30 \circ$$, then the length of $$A D$$ is             .

Answer: 6 or 12


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A B = 2 A C = 4$$, $$C O$$ is the median to the hypotenuse, point P is a moving point on segment $$A O$$, rotate segment $$P C$$ counterclockwise by $$90 \circ$$ about point P to obtain segment $$P Q$$, connect $$C Q$$ and $$O Q$$, when $$P C$$ is perpendicular to one side of $$\triangle A B C$$, the value of segment $$O Q$$ is          .

Answer: $$\sqrt{3} - 1$$ or $$\sqrt{6} - \sqrt{2}$$


As shown in the figure, in the isosceles triangle $$\triangle A B C$$, the base $$B C = 5$$, D is a point on the leg $$A B$$, and $$C D = 4$$, $$B D = 3$$, then the length of $$A D$$ is       .

Answer: $$\frac{7}{6}$$/$$1 \frac{1}{6}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$\angle C = 120^\circ$$, $$A B = 2$$, $$A D = 2 A B$$, points H and G are moving points on sides $$D C$$ and $$B C$$ respectively. Connect $$A H$$ and $$H G$$. Point E is the midpoint of $$A H$$, and point F is the midpoint of $$G H$$. Connect $$E F$$. What is the minimum value of $$E F$$?

Answer: $$\frac{\sqrt{3}}{2}$$/$$0 . 5 \sqrt{3}$$/$$\frac{1}{2} \sqrt{3}$$


As shown in the figure, BE is the median of △ABC, point D is a point on side BC, BD = 2CD, BE and AD intersect at point F. If the area of △ABC is 24, then S△BDF - S△AEF equals     .

Answer: 4


As shown in the figure, the side length of the inscribed regular hexagon $ABCDEF$ in circle $\bigodot O$ is 4, and H is the midpoint of side $AF$. The area of the shaded region in the figure is ______.

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


As shown in the figure, given that the chord $$AB = 6$$ of $$\bigodot O$$, construct a square $$ABCD$$ with $$AB$$ as one side, and side $$CD$$ is tangent to $$\bigodot O$$ at point $$E$$. Then the radius of $$\bigodot O$$ is

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


As shown in the figure, in the isosceles triangle $$A B C$$, $$A B = A C = \sqrt{5}, B C = 2, A D$$ bisects $$\angle B A C, G E$$ is the perpendicular bisector of $$A C$$ intersecting $$A D$$ at point $$F$$, then the length of $$A F$$ is      .

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


As shown in the figure, in parallelogram $$\square A B C D$$, point E lies on $$A D$$, and $$E C$$ bisects $$\angle B E D$$. If $$\angle E B C = 30^\circ$$ and $$B E = 10$$, then the area of $$\square A B C D$$ is        .

Answer: 50


As shown in the figure, in quadrilateral ABCD, ∠ADC = 90°, AC ⊥ BC, ∠ABC = 45°, AC intersects BD at point E. If AB = $$2 \sqrt{10}$$, CD = 2, then the area of △ABE is         .

Answer: $$\frac{60}{7}$$


As shown in the figure, given that the side length of square ABCD is 5, points E and F lie on AD and DC respectively, with AE = DF = 2. BE intersects AF at point G. Point H is the midpoint of BF. Connect GH; then the length of GH is       .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, point O lies on side $$A C$$, and a circle with center O and radius 3 passes exactly through point C and is tangent to side $$A B$$ at point D, intersecting side $$B C$$ at point E. The length of the minor arc $$D E$$ is       (answer in terms of π).

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


As shown in the figure, $$\triangle A B C$$ is inscribed in a circle $$\bigodot O$$ with radius $$\frac{7 \sqrt{3}}{3}$$, $$B D \bot A C$$ at point $$D$$, the extension of $$B D$$ intersects $$\bigodot O$$ at point $$E$$, $$M$$ is the midpoint of $$\overset{⌢}{A B E}$$, connect $$O M$$ intersecting $$A C$$ at point $$F$$, if $$A C = 8$$, $$\angle C = 60 \circ$$, $$A B > 2 B C$$, then $$D E =$$      , $$O F =$$         .

Answer: $$\frac{13 \sqrt{3}}{6}$$     $$\frac{2}{3}$$


As shown in the figure, △ABC is an equilateral triangle, AD is the altitude from A to side BC, E is the midpoint of AC, and P is a moving point on AD. When the sum of PC and PE is minimized, the measure of ∠CPE is        °.

Answer: 60


As shown in the figure, the side length of the equilateral triangle ABC is 4, the radius of $$\bigodot C$$ is $$\sqrt{3}$$, and P is a moving point on side AB. Draw the tangent PQ to $$\bigodot C$$ from point P, with Q as the point of tangency. The minimum value of PQ is        .

Answer: 3


As shown in the figure, in rectangle $$A B C D$$, point E lies on $$A B$$, connect $$C E$$ and $$D E$$, draw $$D F \bot C E$$ through point D, with foot at F. If $$C E = C D$$, $$D F = 3$$, $$B E = 4$$, then $$D E =$$        .

Answer: $$\sqrt{10}$$


As shown in the figure, in $$R t \triangle A O B$$, $$O B = 2 \sqrt{3}, \angle A = 30 \circ, \bigodot O$$ has a radius of $$1,$$ point $$P$$ is a moving point on side $$A B$$, and from point $$P$$, draw a tangent $$P Q$$ to $$\bigodot O$$ (with point $$Q$$ as the point of tangency). Then the minimum length of segment $$P Q$$ is    .

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


In $$\text{Rt} \triangle A B C$$, $$\angle B = 90 \circ$$, $$A B = 6$$, $$\sin C = \frac{3}{5}$$, D is a moving point on side $$A B$$, rotate $$D A$$ about point D so that point A lands at point E on side $$A C$$, draw $$E F \bot D E$$ intersecting side $$B C$$ at point F, connect $$D F$$, when $$\triangle D E F$$ is an isosceles triangle, the length of segment $$C F$$ is      .

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


As shown in the figure, square $$A B C D$$, E and F are the midpoints of $$A B$$ and $$B C$$ respectively, $$A F$$ and $$D E$$ intersect at point G, connect $$C G$$, if $$A B = 2$$, then the length of $$C G$$ is      .

Answer: 2


As shown in the figure, in rectangle $$A B C D$$, $$A B = 8$$, $$A D = 10$$, E is a point on $$C D$$, and $$\triangle A D E$$ is folded along line $$A E$$ such that point D lands exactly at point F on side $$B C$$. Then $$C E =$$      .

Answer: 3


As shown in the figure, in the regular hexagon $$A B C D E F$$, $$M$$ and $$N$$ are two points on the diagonal $$B E$$. Add one of the following conditions: ① $$B M = E N$$; ② $$\angle F A N = \angle C D M$$; ③ $$A M = D N$$; ④ $$\angle A M B = \angle D N E$$. The conditions that can make quadrilateral $$A M D N$$ a parallelogram are          (fill in all the serial numbers of the conditions that meet the requirements).

Answer: ①②④


As shown in the figure, point E lies on the diagonal AC of square ABCD, EF ⊥ AB at point F, connect DE and extend it to intersect side BC at point M and side AB extended at point G. If AF = 2, FB = 1, then MG =          .

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


As shown in the figure, in $$\text{Rt}\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A B = 5$$, $$A C = 10$$, $$D$$ is the midpoint of $$B C$$, $$E$$ is the midpoint of $$A C$$, connect $$B E$$ intersecting $$A D$$ at point $$F$$, then the area of $$\triangle A B F$$ is     .

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


As shown in the figure, in parallelogram $$ABCD$$, $$AB = 4$$, the angle bisector of $$\angle BAD$$ intersects the extension of $$BC$$ at point E and intersects $$DC$$ at point F, and point F is the midpoint of side $$DC$$. $$DG \bot AE$$, with foot at G. If $$DG = 1$$, then the length of $$AE$$ is          .

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


In $$\triangle A B C$$, $$\angle A = 65^\circ$$. Fold $$\angle B$$ and $$\angle C$$ as shown in the figure. If $$\angle A D B' = 35^\circ$$, then $$\angle 1 + \angle 2 + \angle 3 =$$     $$^\circ$$.

Answer: $$265 \circ$$


Place a protractor and an unmarked transparent straightedge as shown in the figure, with the edges of the straightedge intersecting the protractor at points A, B, C, D, where points C and D correspond to the markings of 120 and 60 on the protractor. If the diameter EF of the protractor is 8 cm, then the distance from point O to CD is           cm.

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


As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, with diagonals $$A C$$ and $$B D$$ intersecting at point $$O$$, $$\angle B A C = 90 \circ$$, $$A H \bot B D$$ at point $$H$$, $$A B = 2$$, $$B C = 2 \sqrt{3}$$, then the length of $$A H$$ is        .

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


As shown in the figure, fold the rectangular paper $$A B C D$$ so that $$A D$$ coincides with $$B C$$, obtaining the crease $$E F$$. Flatten the paper again, and fold it once more so that the image point $$A^{'}$$ of point A lies on $$E F$$, and the crease passes through point B, obtaining the crease $$B M$$. Connect $$M F$$. If $$M F \bot B M$$ and $$A B = 6 \text{cm}$$, then the length of $$A D$$ is            $$\text{cm}$$.

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


As shown in the figure, in rectangle ABCD, AB = 2, BC = 1. Rotate rectangle ABCD clockwise 90° about vertex C to obtain rectangle EFCG. Connect AE, take the midpoint H of AE, and connect DH. Then $$D H =$$       .

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


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, chord $$CD \bot AB$$ at point $$E$$, and $$AC = CD$$. If $$AC = 2\sqrt{3}$$, then the radius of $$\bigodot O$$ is      .

Answer: 2


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A B = 10$$, $$B C = 8$$, $$A D$$ is the angle bisector of $$\triangle A B C$$. $$M$$ is a moving point on side $$A C$$, and $$N$$ is a moving point on segment $$A D$$. Connect $$B M$$, $$C N$$, and $$M N$$. When $$C N + M N$$ is minimized, the area of $$\triangle A B M$$ is     .

Answer: $$\frac{72}{5}$$


As shown in the figure, the diagonals $$AC$$ and $$BD$$ of quadrilateral $$ABCD$$ intersect at point $$O$$, with $$OA = OB = OC = OD$$. Draw $$OE \bot BD$$ intersecting $$BC$$ at point $$E$$. If $$AB = 5$$ and $$BE = 7$$, then the length of $$CE$$ is       .

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


As shown in the figure, $$\triangle A B C$$ is inscribed in $$\bigodot O$$ and $$\angle A C B = 90 \circ$$, chord $$C D$$ bisects $$\angle A C B$$, and connect $$A D$$, $$B D$$. If $$A B = 5$$, $$A C = 4$$, then $$B D =$$     , $$C D =$$        .

Answer: $$\frac{5}{2} \sqrt{2}$$     $$\frac{7}{2} \sqrt{2}$$


As shown in the figure, in the acute triangle $$\triangle ABC$$, $$AC = 7$$, the area of $$\triangle ABC$$ is 21, and the angle bisector of $$\angle BAC$$ intersects $$BC$$ at point D. M and N are moving points on $$AD$$ and $$AB$$, respectively. The minimum value of $$BM + MN$$ is        .

Answer: 6


As shown in the figure, given $$AD = AE$$, please add one condition such that $$\triangle ADC \sim \triangle AEB$$. The condition you add is     . (Do not add any letters or auxiliary lines)

Answer: $$A B = A C$$ or $$\angle A D C = \angle A E B$$ or $$\angle A B E = \angle A C D$$.


As shown in the figure, the diagonals $$AF$$ and $$HD$$ of the regular octagon $$ABCDEFGH$$ intersect at point $$M$$. Then the measure of $$\angle AMH$$ is        °.

Answer: 67.5


In the isosceles right triangle $$\triangle A B C$$, $$A C = B C$$, $$D$$ is a point on side $$A B$$, and $$\frac{B D}{A D} = \frac{1}{3}$$. Connect $$C D$$, and construct $$\text{Rt} \triangle D E C$$ downward with $$C D$$ as the hypotenuse. If $$\angle E D C = 30^\circ$$ and $$B C = 2$$, then the length of $$D E$$ is     .

Answer: $$\frac{\sqrt{30}}{4}$$


As shown in the figure, rotate $$\triangle A B C$$ about point A to the position of $$\triangle A E F$$, with point E on side $$B C$$, and $$E F$$ intersecting $$A C$$ at point G. If $$\angle B = 70 \circ$$, $$\angle C = 25 \circ$$, then $$\angle F G C =$$        .

Answer: $$65 \circ$$


As shown in the figure, in $$R t \triangle A B C$$, $$\angle A C B = 90 \circ, A C = 8, B C = 6$$, point P is a moving point in the plane such that $$A P = 3$$, and Q is the midpoint of $$B P$$. During the motion of point P, let the length of segment $$C Q$$ be m. Then the range of m is          .

Answer: $$\frac{7}{2}$$≤m≤$$\frac{13}{2}$$


As shown in the figure, in △ABC, AB = BC, ∠ABC = 90°, F is a point on the extension of AB, and point E is on BC such that AE = CF. If ∠BAE = 25°, then ∠ACF =          degrees.

Answer: 70


As shown in the figure, in square $$ABCD$$, point $$M$$ is a point on $$AB$$. From point $$D$$, draw $$DN \bot DM$$, intersecting the extension of $$BC$$ at point $$N$$. Connect $$MN$$, and let point $$E$$ be the midpoint of $$MN$$. Connect $$BE$$. If $$AM = 1$$ and $$AD = 3$$, then the length of segment $$BE$$ is          .

Answer: $$\sqrt{5}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle B = 90 \circ$$, $$A B = 2$$, $$C D = 8$$. Connect $$A C$$, $$A C \bot C D$$, if $$s i n \angle A C B = \frac{1}{3}$$, then the length of $$A D$$ is         .

Answer: 10


As shown in the figure, the equilateral triangle $$\triangle A B C$$ is inscribed in $$\bigodot O$$. If the radius of $$\bigodot O$$ is $$4$$, then the area of the shaded region is      .

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


As shown in the figure, in the equilateral triangle $$\triangle ABC$$, $$AB = 6$$. With D, the midpoint of $$BC$$, as the center and $$BD$$ as the radius, draw an arc intersecting $$AB$$ and $$AC$$ at points E and F, respectively. Then, with point A as the center and $$AE$$ as the radius, draw another arc. The area of the shaded region in the figure is       .

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


As shown in the figure, $$\triangle A B C$$ is inscribed in $$\bigodot O$$, $$B D$$ is the diameter of $$\bigodot O$$, $$A B = A C$$, $$\angle A = 70 \circ$$, then $$\angle A B D - \angle C B D =$$        $$\circ$$.

Answer: 15


As shown in the figure, the side length of square $$A B C D$$ is 4, point E lies on $$A B$$ such that $$A E = 1$$, and P is a moving point on diagonal $$B D$$. The minimum value of the perimeter of $$\triangle A E P$$ is       .

Answer: 6


As shown in the figure, the side length of square $$A B C D$$ is 2, point E is the midpoint of $$B C$$, $$A E$$ intersects $$B D$$ at point P, F is a point on $$C D$$, connecting $$A F$$ intersects $$B D$$ and $$D E$$ at points M and N respectively, and $$A F \bot D E$$, connecting $$P N$$,

then (I) the length of $$A E$$ is         .
(II) the length of $$P N$$ is         .

Answer: $$\sqrt{5}$$     $$\frac{2 \sqrt{65}}{15}$$


As shown in the figure, let $$G$$ be the centroid of $$\Delta A B C$$. Draw a line through point $$G$$ parallel to $$B C$$, intersecting sides $$A B$$ and $$A C$$ at points $$D$$ and $$E$$, respectively. Let $$\overset{\rightarrow}{G B} = \overset{\rightarrow}{a}$$, $$\overset{\rightarrow}{G C} = \overset{\rightarrow}{b}$$. Express the vector $$\overset{\rightarrow}{D E}$$ in the form $$x \overset{\rightarrow}{a} + y \overset{\rightarrow}{b}$$ (where $$x$$ and $$y$$ are real numbers): _______________.

Answer: $$- \frac{2}{3} \overset{\rightarrow}{a} + \frac{2}{3} \overset{\rightarrow}{b}$$


As shown in the figure, in $$\triangle ABC$$, $$\angle ACB = 90^\circ$$, $$AC = 4$$, $$BC = 3$$. Rotate $$\triangle ABC$$ about point $$B$$ to the position of $$\triangle DBE$$, where point $$D$$ corresponds to point $$A$$, and point $$E$$ corresponds to point $$C$$. If the area of the shaded region in the figure is 4.5, then the tangent of $$\angle CBE$$ is          .

Answer: $$\frac{9}{13}$$


As shown in the figure, in $$\triangle A B C$$, points $$D$$ and $$E$$ are the midpoints of sides $$A B$$ and $$B C$$, respectively. Point $$F$$ lies on the extension of segment $$D E$$, and $$\angle B F C = 90 \circ$$. If $$A C = 4$$ and $$B C = 8$$, then the length of $$D F$$ is      .

Answer: 6


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$\angle A = 30 \circ$$, $$B C = 2$$, point $$D$$ is the midpoint of $$A C$$, point $$E$$ is a moving point on the hypotenuse $$A B$$, fold $$\triangle A D E$$ along the line $$D E$$ to the position $$\triangle A^{'} D E$$, $$A^{'} D$$ intersects $$A B$$ at point $$F$$, if $$\triangle B A^{'} F$$ is a right triangle, then the length of $$A E$$ is        .

Answer: $$1$$ or $$\frac{6}{5}$$


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

Answer: $$\sqrt{10}$$


As shown in the figure, in the equilateral triangle $$\triangle ABC$$ with side length $$a$$, $$BF$$ is the median on side $$AC$$ and $$BF = b$$. Point $$D$$ lies on $$BF$$, and connect $$AD$$. Construct an equilateral triangle $$\triangle ADE$$ on the right side of $$AD$$, and connect $$EF$$. The minimum perimeter of $$\triangle AEF$$ is   , and at this time $$\angle CFE =$$      .

Answer: $$\frac{1}{2} a + b$$     $$90 \circ$$


As shown in the figure, the side length of rhombus ABCD is 6, ∠ABC = 120°, M is a trisection point of side BC, and P is a moving point on diagonal AC. When PB + PM is minimized, the length of PM is ______.

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


As shown in the rhombus $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, with $$A C = 8$$ and $$B D = 6$$. Points E and F lie on sides $$A B$$ and $$C D$$ respectively (point E does not coincide with A or B). Moreover, $$D E \parallel B F$$, and $$D E$$ and $$B F$$ intersect $$A C$$ at points P and Q respectively. Connect $$B P$$ and $$D Q$$. The following four conclusions are given: ① $$A C$$ bisects the perimeter of quadrilateral $$B E D F$$; ② quadrilateral $$B E D F$$ is a rectangle; ③ $$B D$$ bisects $$\angle P D Q$$; ④ when $$D E \bot A B$$, $$\frac{A E}{E D} = \frac{7}{24}$$. Among the above conclusions, the serial numbers of all correct conclusions are           .

Answer: ①③④


As shown in the figure, in $$\triangle A B C$$, $$\angle A = \alpha$$, 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 $$A_{2}$$; $$\hdots$$; the angle bisectors of $$\angle A_{2019} B C$$ and $$\angle A_{2019} C D$$ intersect at point $$A_{2020}$$, yielding $$\angle A_{2020}$$, then $$\angle A_{2020} =$$      .

Answer: $$\frac{\alpha}{2^{2020}}$$


As shown in the figure, in the equilateral triangle $$ABC$$ with side length 6, point D lies on $$BC$$ such that $$BD = 4$$, and $$CQ$$ is the angle bisector of the exterior angle $$\angle ACP$$ of $$\triangle ABC$$. Reflecting $$\triangle ABD$$ over $$AD$$ yields $$\triangle AED$$, and $$DE$$ intersects $$CQ$$ at point F. Find the length of $$CF$$.

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


As shown in the figure, the rhombus $$A B C D$$ has side length 4, $$\angle A B C = 60 \circ$$, points E and F are moving points on sides $$A D$$ and $$C D$$ respectively, and $$\angle B E F = 120 \circ$$. What is the maximum length of segment $$D F$$?

Answer: $$12 - 8 \sqrt{2}$$


As shown in the figure, in square ABCD, AB = 6, E is the midpoint of side CD, and F is a moving point on segment BC. Fold triangle ECF along the line EF to obtain triangle EC'F, and connect AC'. The minimum value of AC' is ______.

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


As shown in the figure, if point O is the midpoint of diagonal BD of rectangle ABCD, fold it as shown in the figure such that side AB falls onto BD and side CD also falls onto BD, with points A and C coinciding exactly at point O. Connect EC, intersecting BD at point G and DF at point H.

(1) The measure of angle AEB is ______ degrees; (2) The value of GH/HC is ______.

Answer: $$60$$     $$\frac{4}{5}$$


As shown in the figure, with the vertices $A$, $B$, and $C$ of the equilateral triangle $\triangle ABC$ as centers, and with $AB$ as the radius, arcs are drawn. The closed figure formed by these three arcs is called a Reuleaux triangle. If the perimeter of the Reuleaux triangle is $2\pi$, then the area of the Reuleaux triangle is          .

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


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, points E and F are on sides $$AD$$ and $$AB$$ respectively, and $$DE = BF$$. Connect $$BE$$ and $$CF$$. Then the minimum value of $$BE + CF$$ is           .

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


As shown in the figure, in quadrilateral $$A C B D$$, diagonals $$A B$$, $$C D$$ intersect at point O, $$\angle A C B = 90 \circ$$, $$B D = C D = 10$$, $$B C = 16$$, if $$\angle D A B = 2 \angle A B C$$, then the value of $$\frac{A D}{A B}$$ is                   .

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


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, point O is the midpoint of diagonal $$AC$$, point Q is a moving point on segment $$OA$$ (point Q does not coincide with point O or A). Connect $$BQ$$ and extend it to intersect side $$AD$$ at point E. From point Q, draw $$FQ \bot BQ$$ intersecting $$CD$$ at point F. Connect $$BF$$ and $$EF$$. $$BF$$ intersects diagonal $$AC$$ at point G. From point C, draw $$CH \parallel QF$$ intersecting $$BE$$ at point H. Connect $$AH$$. The following four conclusions: ① $$BQ = QF$$; ② The perimeter of $$\triangle DEF$$ is 8; ③ $$\angle BQG = \angle BEF$$; ④ The minimum value of segment $$AH$$ is $$2\sqrt{5} - 2$$. Among these, the correct conclusions are      . (Fill in the serial numbers)

Answer: ①②④


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle A C B = 54 \circ, \angle B A C = 64 \circ$$, diagonal $$B D$$ bisects $$\angle A B C$$, and $$\angle B C D + \angle D C A = 180 \circ$$. Then $$\angle A D C$$ is          degrees.

Answer: 59


As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ intersects $$B D$$ at point O, and $$A C \bot B D$$, $$A B = A C = B D$$. Point E is the intersection of the angle bisectors of $$\angle A B D$$ and $$\angle C A B$$. Connect $$E D$$ and $$E C$$. Then
(1) $$\angle A E B =$$      ;
(2) Among the following conclusions: ① $$A D = D C$$; ② $$\angle A C E = \angle A B E$$; ③ $$B E \bot E C$$; ④ $$S_{ \triangle A E B} = S_{\triangle E D C}$$; the correct ones are      .

Answer: $$135 \circ$$     ②③④


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle A = 30 \circ$$, $$B C = 4$$, and $$D$$ is a moving point on $$A B$$. Construct an isosceles $$\text{Rt} \triangle D C E$$ with $$D C$$ as the hypotenuse to the right side such that $$\angle C E D = 90 \circ$$. Connect $$B E$$. What is the minimum value of segment $$B E$$?

Answer: $$\sqrt{2}$$


As shown in the figure, $$BD$$ is the angle bisector of the isosceles $$\triangle ABC$$, $$AB = AC = 6$$, $$BC = 8$$, $$E$$ is a moving point on segment $$BD$$ (excluding endpoints), connect $$AE$$, construct $$\angle EAF = \angle BAC$$, and $$AE = AF$$, connect $$DF$$. When the perimeter of $$\triangle ADF$$ is minimized, the value of $$\frac{AF}{DF}$$ is      .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle C A B = 40 \circ$$. Now fold $$\triangle A B C$$ along a straight line passing through point $$A$$, such that point $$B$$ lands at point $$E$$ on the extension of segment $$A C$$. The angle bisector of $$\angle A C B$$ intersects the fold line at point $$D$$. Connect $$B D$$ and $$D E$$. If one interior angle of $$\triangle C D E$$ is three times another interior angle, then the measure of $$\angle A C B$$ is      .

Answer: $$80 \circ$$ or $$110 \circ$$ or $$\frac{500 \circ}{3}$$.


As shown in the figure, in the rhombus $$A B C D$$, point E is the midpoint of side $$A B$$, and point P moves along side $$B C$$. Fold $$\triangle B E P$$ along the crease $$E P$$ to obtain $$\triangle F E P$$, and connect $$D F$$. If $$A B = 4$$, $$\angle B = \text{60} \circ$$, then $$E F =$$   , and the minimum value of $$D F$$ is          .

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


As shown in the figure, let $$P$$ be a point on side $$BC$$ of parallelogram $$ABCD$$. Fold $$\triangle ABP$$ along line $$AP$$, such that point $$B$$ lands at point $$E$$ inside parallelogram $$ABCD$$, and $$EA = ED$$. If $$AB = 5$$, $$AD = 8$$, and the sine of $$\angle B$$ is $$0.8$$, then the length of $$BP$$ is      .

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


As shown in the figure, in parallelogram $$\square A B C D$$, $$t a n A = \frac{4}{3}$$, $$A D = \frac{55}{4}$$, point E lies on side $$A D$$. Fold $$\triangle A B E$$ along line $$B E$$ such that the image of side $$A B$$, denoted $$A^{'} B$$, is perpendicular to $$D C$$, with foot at F. $$A^{'} E$$ intersects $$D C$$ at point G. When point G is exactly the midpoint of $$A^{'} E$$, the length of $$A B$$ is          .

Answer: 14


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle B = 30 \circ$$, $$B C = 9$$, $$D$$ is a point on $$A C$$ such that $$A D = 2 D C$$, and $$P$$ is a moving point on side $$B C$$. When $$\triangle A P D$$ is a right triangle, the length of $$B P$$ is      .

Answer: 3 or 6 or 7


As shown in the figure, the side length of square $$A B C D$$ is 4, and points $$E$$, $$F$$ lie on sides $$D C$$, $$B C$$ respectively, with $$B F = C E$$, and $$A E$$ bisects $$\angle C A D$$. Connect $$D F$$, which intersects $$A E$$ and $$A C$$ at points $$G$$ and $$M$$ respectively. Point $$P$$ is a moving point on segment $$A G$$. From point $$P$$, draw $$P N \bot A C$$, with foot of perpendicular at $$N$$. Connect $$P M$$. Then the minimum value of $$P M + P N$$ is , and $$S_{\Delta A D M} =$$    .

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


As shown in the figure, in the isosceles right triangle $$\triangle A B C$$, $$\angle C = 90 \circ$$, $$A B = 6 \sqrt{2}$$, fold $$\triangle A B C$$ along a certain straight line such that point $$B$$ lands on the midpoint of $$A C$$. If the crease intersects $$A B$$ at point $$M$$, then the length of $$A M$$ is      .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle A B C = 90 \circ$$, $$A B = 2$$, $$B C = 4$$, point E lies on side BC, and segment AE is rotated clockwise about point E by $$90 \circ$$ to obtain $$F E$$, and $$C F$$ is connected. The minimum value of $$C F$$ is      .

Answer: $$\sqrt{2}$$


Mathematician Fehrbe proposed the idea of using diagrams instead of calculations; such diagrams are called "nomograms." The figure shown is a nomogram illustrating the relationship among x, y, and z. It consists of three rays a, b, and c emanating from point O, each with identical scales, and the endpoint of each ray is marked as 0. Rays a and c, as well as b and c, intersect at an angle of $$60 \circ$$. Points A and B are selected on rays a and b, respectively, corresponding to scale values x and y. Connecting A and B with a straightedge intersects ray c at point C, and the scale value at point C is z.
(1) If $$x = 20$$, $$y = 12$$, then z =        ;
(2) If $$x = 2 y$$, then $$\frac{z}{y} =$$        .

Answer: $$7.5$$     $$\frac{2}{3}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$A C$$ is a diagonal, $$A E \bot B C$$ at point $$E$$, point $$F$$ lies on the extension of $$A E$$ such that $$\angle C A F = \angle C F A$$, and ray $$C F$$ intersects the extension of segment $$A B$$ at point $$G$$. If $$A B = \sqrt{10}$$, $$A D = 4$$, $$\tan \angle A B C = 3$$, then the length of $$B G$$ is      ; if point $$M$$ is the midpoint of $$A G$$, and segment $$M E$$ is connected, then the length of $$M E$$ is      .

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


As shown in the figure, in rectangle ABCD, $$AB = 6$$, $$BC = 5$$, E and F are moving points on sides AB and BC respectively, and $$EF = 4$$, M is the midpoint of EF, and P is a moving point on side AD. Then the minimum value of $$CP + PM$$ is      .

Answer: 11


$$\triangle A B C$$ is an equilateral triangle with side length 5, and $$\triangle D C E$$ is an equilateral triangle with side length 3. The line BD intersects the line AE at point F. As shown in the figure, rotate $$\triangle D C E$$ around point C by one full revolution. During this rotation, the minimum length of segment AF is           .

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


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, and $$E$$ is a moving point on diagonal $$BD$$. An isosceles right triangle $$AEF$$ is constructed to the right with $$AE$$ as the hypotenuse. Then the minimum value of $$FC + FD$$ is      .

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


As shown in the figure, in quadrilateral $$A B C D$$, $$A C$$ and $$B D$$ intersect at point $$E$$, $$A C = B C$$, $$\angle A C B = \angle A D B = 90^\circ$$, $$B D = a$$, $$A D = b$$, express the area of $$\triangle B C D$$ in terms of $$a$$ and $$b$$ as      .

Answer: $$\frac{a^{2} - a b}{4}$$


As shown in the figure, in square $$ABCD$$, point P lies on side $$AD$$, point E is symmetric to point B with respect to line $$CP$$, and ray $$ED$$ intersects the extension of $$CP$$ at point F. If $$AD = 4PD$$ and $$EF = 16\sqrt{2}$$, then the length of $$BC$$ is      .

Answer: $$4 \sqrt{17}$$


As shown in the figure, in $$\triangle A B C$$, $$S_{\triangle A B C} = 10$$, points D and E lie on $$B C$$ and $$A C$$ respectively, $$C D = 2 B D$$, $$C E = 2 A E$$, and $$B E$$ intersects $$A D$$ at point F. Then the area of $$\triangle A F E$$ is        .

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


In $$\triangle A B C$$, with $$B C = 6 \text{cm}, \angle A = 60 \circ$$, the maximum value of $$A B + \frac{\sqrt{3} - 1}{2} A C$$ is        .

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


As shown in the figure, $$AD = 4$$, there is a moving point C on side $$AD$$; construct equilateral $$\triangle ABC$$ and equilateral $$\triangle CDE$$ above side $$AD$$ with sides $$AC$$ and $$CD$$ respectively; connect $$BE$$, take the midpoint $$F$$ of side $$BE$$, and connect $$CF$$; then the minimum value of $$CF$$ is       .

Answer: $$\sqrt{3}$$


As shown in the figure, in $$\triangle A B C$$, point $$M$$ is the midpoint of $$B C$$, $$A D$$ bisects $$\angle B A C$$, and $$B D \bot A D$$ at point $$D$$. Extend $$B D$$ to intersect $$A C$$ at point $$N$$. If $$A B = 12$$, $$A C = 18$$, then $$M D =$$                       .

Answer: $$3$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = 3$$, $$\angle B A C = 120^\circ$$, construct $$\angle A C M = \angle A C B$$ on the other side of $$C A$$ with respect to $$\angle A C B$$. Let point D be any point on side $$B C$$ (excluding endpoints). On ray $$C M$$, take $$C E = B D$$. Connect $$A D$$, $$D E$$, and $$A E$$. Let $$A C$$ intersect $$D E$$ at point F. Then the maximum value of segment $$C F$$ is        .

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


As shown in the figure, AB∥CD, AC∥BD, CE bisects ∠ACD and intersects BD at point E. Point F lies on the extension of CD, and ∠BEF = ∠CEF. If ∠DEF = ∠EDF, then the measure of ∠A is     $$\circ$$.

Answer: 108


As shown in the figure, in the isosceles triangle $$\triangle ABC$$, $$\angle BAC = 150^\circ$$, point $$D$$ lies on $$AB$$ such that $$AD = 1$$ and $$BD = 6$$, and point $$E$$ lies on side $$BC$$. If the point $$F$$, which is the image of point $$E$$ rotated counterclockwise by $$15^\circ$$ about point $$D$$, lies exactly on $$AC$$, then the length of $$BE$$ is            .

Answer: $$1 + 6 \sqrt{2}$$


In quadrilateral $$A B C D$$, $$E$$, $$F$$ are the midpoints of sides $$A B$$, $$A D$$ respectively. If $$B C = 15$$, $$C D = 9$$, $$E F = 6$$, $$\angle A F E = 55^\circ$$, then $$\angle A D C =$$      .

Answer: 145°


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, point $$D$$ is a moving point on side $$A C$$, connect $$B D$$, rotate segment $$B D$$ clockwise around point $$B$$ by $$60 \circ$$ to obtain segment $$B E$$, connect $$C E$$. If $$A B = 2 \sqrt{10}$$, $$B C = 3 A B$$, then the minimum value of segment $$C E$$ is      .

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


As shown in the figure, in $$\triangle A B C$$ and $$\triangle A D E$$, $$\angle B A C = \angle D A E = 90 \circ$$, $$A B = A C = 2 A D = 2 A E = 4$$, point O lies on side $$B C$$ such that $$O C = 3 O B$$. Rotate $$\triangle A D E$$ clockwise about point A, connect $$C E$$, and let P be the midpoint of $$C E$$. Then the maximum value of $$O P$$ is          .

Answer: $$\sqrt{10} + 1$$


As shown in the figure, point P is inside ∠AOB, and points M and N are moving points on sides OA and OB, respectively, and points M and N do not coincide with point O.

(1) If point P is moved to a position inside ∠AOB such that OP bisects ∠AOB, and PN ∥ OA, ON = 2, then the length of PN is ______; (2) If ∠AOB = 60°, OP = a, and as points M and N move, when the perimeter of △PMN is minimized, the distance from point O to line MN is ______. (Expressed as an algebraic expression in terms of a)

Answer: 2     $$\frac{1}{2} a$$


As shown in the figure, in the rhombus $$ABCD$$ with side length 4, $$\angle ABC = 120^\circ$$. Translate $$\triangle ADC$$ along the ray $$AC$$ to obtain $$\triangle A'D'C'$$. Connect $$A'B$$ and $$D'B$$. The minimum value of $$A'B + D'B$$ is      .

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


Given: As shown in the figure, the area of the equilateral triangle $$A B C$$ is $$3 \sqrt{3}$$, and $$D$$, $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, with $$A D = C E$$. Then the minimum value of $$D E$$ is      .

Answer: $$\sqrt{3}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 8$$, points E and F lie on sides $$A D$$ and $$B C$$ respectively, and $$A E = 3$$. Folding along line $$E F$$, the image of point A, denoted $$A^{'}$$, falls exactly on diagonal $$A C$$, and the image of point B is $$B^{'}$$. Point M is a moving point on segment $$A A^{'}$$. Find the minimum value of $$E M + \frac{\sqrt{5}}{5} A^{'} M$$        .

Answer: $$\frac{12}{5}$$/$$2 . 4$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C = 2$$, point $$D$$ is a moving point on side $$B C$$ (not coinciding with points $$B$$ or $$C$$), $$C E$$ is perpendicular to $$A D$$, intersecting $$A B$$ at point $$E$$, with foot of perpendicular at point $$H$$, connect $$B H$$ and extend it to intersect $$A C$$ at point $$F$$,
① If $$A D$$ is the median of side $$B C$$, then $$D H = \frac{\sqrt{5}}{5}$$;
② If $$A D$$ bisects $$\angle C A B$$, then $$\frac{C D}{B D} = \frac{\sqrt{2}}{2}$$;
③ If $$B D = 2 C D$$, then $$A E = 2 B E$$;
④ The minimum value of $$B H$$ is $$\sqrt{5} - 1$$.
The correct sequence numbers above are      .

Answer: ①②④


In quadrilateral $$A B D C$$, $$A C = 3$$, $$A B = 5$$, $$B D = C D$$, $$\angle B D C =\text{90} \circ$$, then the maximum value of $$A D$$ is      .

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


As shown in the figure, in square $$ABCD$$, $$AB = 8$$, and there is a moving point P on diagonal $$AC$$ (point P does not coincide with point C or point A). Construct square $$DPFG$$ with $$DP$$ as a side. If E is the midpoint of $$DC$$, connect $$EG$$, then the minimum value of $$EG$$ is          .

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


As shown in the figure, the equilateral triangle $$\triangle A B C$$ has side length $$4$$, and $$D$$, $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, with $$B D = A E$$. Let $$F$$ be the midpoint of $$D E$$, and connect $$A F$$. When $$A F = \frac{\sqrt{13}}{2}$$, the length of $$B D$$ is     .

Answer: $$1$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, points D and E lie on $$A C$$ and $$B C$$ respectively, and $$\angle C D E = \angle B$$. When $$\triangle C D E$$ is folded along $$D E$$, point C lands exactly at point F on side $$A B$$. If $$A C = 8$$ and $$B C = 6$$, then the length of $$C D$$ is

Answer: $$\frac{25}{8}$$


In rhombus ABCD, ∠D = 60°, CD = 4, E is a point inside the rhombus such that AE = 2. Connect CE, let F be the midpoint of CE, connect BF, let G be the midpoint of BF, connect AG. Then the maximum value of AG is     .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4, A D = 6$$, points E and F lie on sides $$A B$$ and $$C D$$ respectively, point M is a moving point on segment $$E F$$, and from point M, a perpendicular to $$E F$$ intersects sides $$A D$$ and $$B C$$ at points G and H respectively. If segment $$E F$$ exactly bisects the area of rectangle $$A B C D$$, and $$D F = 1$$, then the length of $$G H$$ is      .

Answer: $$\frac{4}{3} \sqrt{10}$$


As shown in the figure, quadrilateral $$A B C D$$ and quadrilateral $$C E F G$$ are both squares. Point E is a moving point on the extension of $$D C$$, and point G lies on ray $$C B$$ (not coinciding with point C). H is the midpoint of $$D F$$, and connect $$G H$$. If $$A D = 8$$, then the minimum value of $$G H$$ is       .

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


As shown in the figure, in quadrilateral $$A B C D$$, $$A B = 6$$, $$A D = 2$$, the area of quadrilateral $$A B C D$$ is $$10 \sqrt{2}$$, connect diagonals $$A C$$, $$B D$$, where $$B D \bot A D$$, then the minimum value of $$A C + B C$$ is       .

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


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle with side length $$5$$, point $$D$$ is a point outside $$\triangle A B C$$, $$\angle B A D > \angle B A C$$, $$A D = A C$$. If $$C D = 6$$, connect $$B D$$, then the length of segment $$B D$$ is           .

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


As shown in the figure, in right triangle ABC, ∠ACB = 90°, AC = 4, BC = 3, point D is the midpoint of AC. Rotate CD counterclockwise around point C; during the rotation, the corresponding point of D is point E. Connect AE and BE. The minimum area of triangle AEB is       .

Answer: 1


As shown in the figure, point $$P$$ is a moving point on the diagonal $$BD$$ of square $$ABCD$$, $$PE \bot BC$$ at point $$E$$, $$PF \bot CD$$ at point $$F$$, and connect $$EF$$. The following 5 conclusions are given: ① $$AP = EF$$; ② $$AP \bot EF$$; ③ $$\triangle APD$$ is always an isosceles triangle; ④ $$\angle PFE = \angle BAP$$; ⑤ The minimum value of $$EF$$ equals $$\frac{1}{2} BD$$. The correct conclusion(s) is/are      .

Answer: ①②④⑤


As shown in the figure, in $$\triangle A B C$$, $$A C = B C$$, $$\angle A C B = 90 \circ$$, $$A E$$ bisects $$\angle B A C$$ and intersects $$B C$$ at point $$E$$, $$B D \bot A E$$ intersects the extension of $$A E$$ at point $$D$$, $$D M \bot A C$$ intersects the extension of $$A C$$ at point $$M$$, connect $$C D$$. The following conclusions: ① $$\angle C D A = 45 \circ$$; ② $$C D = \frac{1}{2} A E$$; ③ $$A C + C E = A D$$; ④ $$\frac{A C + A B}{A M}$$ is a constant. The correct conclusions are      . (Fill in the serial numbers)

Answer: ①②④


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, $$A B = 4$$, $$A D \parallel B C$$, $$A D = \frac{1}{2} A B$$, point $$E$$ is a moving point on $$A C$$, connect $$D E$$, point $$F$$ is the midpoint of $$D E$$, connect $$B F$$, then the minimum value of $$B F$$ is      .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 3 \sqrt{3}$$, $$P$$ is a point on $$B C$$, and an equilateral triangle $$\triangle A P Q$$ is constructed with $$A P$$ as a side (with points $$A$$, $$P$$, $$Q$$ arranged counterclockwise). Connect $$C Q$$ and $$D Q$$. Then the minimum value of $$C Q + D Q$$ is      .

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


As shown in the figure, in square $$ABCD$$, point E is a point on side $$AD$$, and point F lies on the extension of side $$DC$$, with $$AE = CF$$. Connect $$EF$$ and let it intersect diagonal $$AC$$ at point G. If $$AB = 8$$ and $$AE = 2$$, then the length of segment $$DG$$ is      .

Answer: $$\sqrt{34}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A B C = 90 \circ$$, $$B A = B C = 4$$, point D is the midpoint of side $$B C$$. Connect $$A D$$, draw $$B E \bot A D$$ through point B, intersecting at point E, and extend $$B E$$ to intersect $$A C$$ at point F. Then the length of $$E F$$ is          .

Answer: $$\frac{8 \sqrt{5}}{15}$$/$$\frac{8}{15} \sqrt{5}$$


In the isosceles triangle $$A B C$$, $$A C = B C = 4$$, $$\angle A C B = 90 \circ$$, $$D$$ is any point in the plane, connect $$A D$$, when $$A D = 2$$, rotate $$A D$$ clockwise 90° about point $$A$$ to obtain $$A E$$, connect $$D E, B D, B E$$, take the midpoint $$M$$ of $$B D$$, connect $$C M$$. When $$B, M, E$$ are collinear, the length of $$C M$$ is           .

Answer: $$\frac{\sqrt{30} + \sqrt{2}}{2}$$ or $$\frac{\sqrt{30} - \sqrt{2}}{2}$$.


As shown in the figure, in the rhombus $$A B C D$$, $$B C = 4$$, $$\angle A B C = 60^{\circ}$$, on side $$B C$$ there is a segment $$E F$$ moving from $$B$$ to $$C$$, stopping when point $$F$$ reaches point $$C$$, with point $$E$$ to the left of point $$F$$, $$E F = 1$$, connect $$A E$$ and $$A F$$, then the minimum perimeter of $$\triangle A E F$$ is      .

Answer: $$8$$


As shown in the figure, in parallelogram $$A B C D$$, point E lies on $$A D$$, $$B C = 3 E D$$, connect $$B E$$, $$D F \bot A B$$ at point F, intersecting $$B E$$ at point G, $$\angle A B E = 2 \angle A D F$$, $$B G = G D$$, if $$C D = 5$$, then the length of segment $$B F$$ is           .

Answer: 2


As shown in the figure, in $$\triangle A B C$$, quadrilateral $$E B C F$$ is folded along $$E F$$, causing point $$B$$ and point $$C$$ to land at points $$B^{'}$$ and $$C^{'}$$ respectively. Then, $$\triangle A E F$$ is folded along $$A F$$, causing point $$E$$ to land at point $$E^{'}$$. If $$\angle B A C = 50 \circ$$ and $$\angle 1 = 65 \circ$$, then the measure of $$\angle 3$$ is      .

Answer: $$85 \circ$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A = 90 \circ$$, $$A B = A C$$, $$A D = 4$$, $$A B = 10$$, points $$D$$, $$E$$ lie on sides $$A B$$, $$A C$$ respectively, $$A D = A E$$, connect $$D C$$, points $$M$$, $$P$$, $$N$$ are the midpoints of $$D E$$, $$D C$$, $$B C$$ respectively.

(1) The area of $$\triangle P M N$$ is     , (2) When $$\triangle A D E$$ is rotated freely about point $$A$$ in the plane, the maximum area of $$\triangle P M N$$ is        .

Answer: $$\frac{9}{2}$$     $$\frac{49}{2}$$


Through studying the textbook section "13.4 Shortest Path Problem", we have experienced the role of axial symmetry transformations. Please use your knowledge of axial symmetry to solve the following problem: As shown in the figure, $$C$$ is the midpoint of $$AB$$, $$\angle ACD + \angle BCE = 60^\circ$$, $$AD = 2$$, $$BE = 4.5$$, $$AB = 6$$, then the maximum value of $$DE$$ is      .

Answer: 9.5


As shown in the figure, in a set of triangle rulers, the longer leg of the triangle containing the $$30^\circ$$ angle ($$\triangle ABC$$) coincides with the hypotenuse of the triangle containing the $$45^\circ$$ angle ($$\triangle ACD$$). Points P and Q lie on sides $$AC$$ and $$BC$$, respectively, $$AB$$ intersects $$CD$$ at E, and quadrilateral $$EPQB$$ is a parallelogram with area 3. Find the length of segment $$DE$$.

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


If a rhombus has one diagonal equal to its side length, it is called a "perfect rhombus." As shown in the figure, given that the "perfect rhombus" ABCD has side length 4, BD is its shorter diagonal, and points M and N are two moving points on sides AD and CD respectively, satisfying AM + CN = 4. Let the area of triangle BMN be S. Then the range of S is     .

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


(1) If the polynomial $$m x^{3} + n x^{2} - 61 x - 36$$ is divisible by $$2 x + 1$$ and $$3 x - 4$$, then $$m - n =$$_________.
(2) As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = \angle B C A = 44 \circ$$, $$M$$ is a point inside $$\triangle A B C$$ such that $$\angle M C A = 30 \circ$$, $$\angle M A C = 16 \circ$$, then the measure of $$\angle B M C$$ is_________.

Answer: (1)$$- 10$$;(2)$$150 \circ$$


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, and $$C$$ is a moving point on $$\bigodot O$$. Construct a square $$ACEF$$ on the left side of $$AC$$. Connect $$BF$$. Then the maximum value of $$\frac{BF}{AB}$$ is         .

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


As shown in the figure, in parallelogram $$A B C D$$, $$A B = 3$$, $$B C = 8$$, $$\angle A B C = 60 \circ$$, $$F$$ is the midpoint of $$B C$$, and $$E$$ is a point on the extension of $$C D$$. If $$A F$$ bisects $$\angle B A E$$, then $$D E =$$       .

Answer: 7


As shown in the figure, in square $$ABCD$$, point E lies on $$AB$$, with $$AE : BE = 2 : 1$$. Connect $$DE$$, and from point A draw $$AG \bot DE$$ intersecting $$DE$$ at point $$G$$. Extend $$AG$$ to intersect $$BC$$ at point $$F$$. Let the ratio of the area of quadrilateral $$BEGF$$ to the area of $$\triangle AEG$$ be $$a$$, and the ratio of the area of square $$ABCD$$ to the area of $$\triangle AEG$$ be $$b$$. Then $$a + b =$$       .

Answer: 12


As shown in the figure, this is a rectangular football field, with $$AB$$ as the goal, $$CD \bot AB$$ at point D, and $$AB = a$$ meters. A player dribbles along $$CD$$ toward the goal $$AB$$ and prepares to shoot from point Q. It is known that $$BD = 3a$$ meters and $$QD = 3a$$ meters. When the goalkeeper spreads his arms, the defensive range is approximately $$0.25a$$ meters. At this moment, the goalkeeper stands within the angle $$\angle AQB$$, with arms spread as $$MN$$ perpendicular to $$AQ$$ for defense. When the midpoint of $$MN$$ is at a distance of _____ meters from $$AB$$, the goalkeeper can just successfully defend.

Answer: $$\frac{37}{20} a$$/$$\frac{37 a}{20}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle B = 90 \circ$$, point $$E$$ lies on $$A B$$, connect $$C E$$, fold $$\triangle B C E$$ along the straight line $$C E$$ to obtain $$\triangle F C E$$, and point $$F$$, the image of point $$B$$, lies exactly on $$A C$$. Draw $$F D \bot A B$$ with foot at point $$D$$, point $$M$$ is a point on the extension of $$D F$$, connect $$C M$$. Point $$N$$ lies on $$C M$$, point $$R$$ lies on $$C F$$, take a point $$Q$$ on the extension of $$C M$$, connect $$F N$$, $$R N$$, and $$R Q$$. If $$\angle C F N = \angle Q$$, $$R N = F R = C R$$, $$F N = N Q$$, $$A B = \frac{40}{3}$$, $$A F = \frac{1}{2} A B$$, then the length of segment $$R Q$$ is      .

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


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = 2 A D = 4$$, $$\angle D = 60^\circ$$, point $$P$$ is a point on side $$C D$$, connect $$P B$$, fold $$\triangle B C P$$ along $$P B$$ so that point $$C$$ lands at point $$N$$, where $$C P \geq 2$$, let $$P N$$ intersect $$A B$$ at point $$M$$, if the area of $$\triangle B M P$$ is $$x$$, then the range of $$x$$ is      .

Answer: $$\sqrt{3} \leq x \leq \frac{7 \sqrt{3}}{5}$$


As shown in the figure, the side length of square $$A B C D$$ is 8, point $$E$$ is a point on side $$B C$$ such that $$B E = 2$$, point $$F$$ is the midpoint of side $$A B$$, connect $$E F$$, and construct an isosceles right triangle $$\text{Rt} \triangle E G F$$ with $$E F$$ as one leg to the right, such that $$\angle E F G = 90^\circ$$, connect $$C G$$, then the length of $$C G$$ is      .

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


As shown in the figure, $$AB$$ is the diameter of semicircle $$O$$, chord $$AC$$ is parallel to radius $$OD$$, connect $$CD$$, $$CD = AO$$, rotate semicircle $$O$$ about point $$A$$, with $$O$$'s corresponding point being $$O^{'}$$, semicircle $$O^{'}$$ intersects $$AB$$ at points $$E$$ and $$D$$, when points $$E$$, $$D$$, and $$B^{'}$$ lie on the same straight line, if $$BB^{'} = 8 \sqrt{2}$$, then the area of quadrilateral $$AODC$$ is      .

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


As shown in the figure, in $$\triangle ABC$$, $$AC = 10$$, $$BC = 14$$, $$\tan \angle ACB = \frac{4}{3}$$, rotate $$\triangle ABC$$ about point C to obtain $$\triangle DEC$$, with points A and B corresponding to points D and E respectively, connect $$AD$$. If points M and N are the midpoints of $$BC$$ and $$AD$$ respectively, connect $$MN$$, then the range of possible lengths of $$MN$$ is           .

Answer: $$4 \sqrt{2} - 5 \leq M N \leq 4 \sqrt{2} + 5$$


As shown in the figure, there is a right triangle paper $$ABC$$ containing a $$30^\circ$$ angle, with the shortest side $$BC = 1$$. Three rhombuses of maximum area are folded out, each using $$\angle A$$, $$\angle B$$, and $$\angle C$$ as one of their interior angles. The sum of the maximum and minimum areas of these three largest rhombuses is      .

Answer: $$42 - \frac{214 \sqrt{3}}{9}$$


As shown in the figure, rectangles $$A B C D$$ and $$A E F G$$, with $$A D = 12$$, $$A B = 9$$, $$A G = 8$$, $$A E = 6$$. Rectangle $$A E F G$$ rotates about point A. The following conclusions are given: ① $$4 B E = 3 D G$$; ② $$B E \bot D G$$; ③ $$D E^{2} + B G^{2} = 315$$; ④ When $$\angle B A G = 60 \circ$$, $$4 S_{\Delta A B G} = 3 \sqrt{3} S_{\Delta A D G}$$. Among these, the correct conclusions are            .

Answer: ①②④


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = 25, B C = 30$$, point $$D$$ is a moving point on side $$A C$$, and $$\triangle B C D$$ is folded along $$B D$$ to obtain $$\triangle B E D$$, with $$B E$$ intersecting $$A C$$ at point $$F$$. Then the maximum value of $$E F$$ is            .

Answer: 6


As shown in the figure, in △ABC (AB > BC), G lies on the extension of CB. The perpendicular bisector of side AC, DE, intersects the angle bisector of ∠ABG at point M, intersects AB at point D, and intersects AC at point E. MN ⊥ AB at N. Given AB = 13, BC = 9, MN = 3, the area of △BMN is    .

Answer: 3


As shown in the figure, $$AD$$ is the altitude of $$\triangle ABC$$, $$\angle BAC = 45^\circ$$. If $$AD = 18$$ and $$DC = 6$$, then the area of $$\triangle ABC$$ is      .

Answer: $$135$$


As shown in the figure, the side length of square $$A B C D$$ is 4. Extend $$C B$$ to $$E$$ such that $$E B = 2$$, and construct square $$E F G B$$ above with $$E B$$ as a side. Extend $$F G$$ to intersect $$D C$$ at $$M$$. Connect $$A M$$ and $$A F$$. Let $$H$$ be the midpoint of $$A D$$, and connect $$F H$$, which intersects $$A B$$ and $$A M$$ at points $$N$$ and $$K$$, respectively. The following conclusions are given: ① $$\triangle A N H \sim \triangle G N F$$; ② $$F K = 3 N K$$; ③ $$\angle A F N = \angle H F G$$; ④ $$S_{\triangle A F N} : S_{\triangle A D M} = 1 : 4$$. Among these, the correct conclusions are     .

Answer: ①②④


As shown in the figure, fold the square ABCD with side length 2 along point A, such that point B lands at $$B^{'}$$, connect D$$B^{'}$$, and let point F be the midpoint of D$$B^{'}$$. Then the minimum value of CF is      .

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


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 $$P$$ is a moving point on $$B E$$, point $$Q$$ is the midpoint of $$P C$$, connect $$A Q$$, then the minimum length of $$A Q$$ is         .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = 3$$, $$B C = 4$$, points D and E are the midpoints of sides $$B C$$ and $$A C$$ respectively, connect $$D E$$, point F is a moving point on side $$A B$$, and $$C F = D E$$, then the length of $$A F$$ is      .

Answer: $$2.5$$ or $$1.1$$


As shown in the figure, in rectangle ABCD, AB = 16, BC = 12, E is the midpoint of side BC, point F lies on side AB, and ∠EDF = 45°. Find the length of AF.

Answer: $$\frac{60}{11}$$


As shown in the figure, in $$\triangle ABC$$, $$\angle A = 30^\circ$$. A sufficiently large right triangle ruler $$PMN$$ ($$\angle M = 90^\circ$$, $$\angle MPN = 30^\circ$$) is placed as shown in the figure, with vertex P sliding along side AC, and the leg $$PM$$ of the triangle ruler always passing through point B, while the hypotenuse $$PN$$ intersects $$AB$$ at point D. If point P slides such that both $$\triangle PAD$$ and $$\triangle PBC$$ are isosceles triangles, then the measure of $$\angle C$$ is        .

Answer: $$30 \circ$$ or $$75 \circ$$ or $$52.5 \circ$$


As shown in the figure, given $$\angle M O N = 30 \circ$$, points $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, … lie on ray ON, and points $$B_{1}$$, $$B_{2}$$, $$B_{3}$$, … lie on ray OM. $$\Delta A_{1} B_{1} A_{2}$$, $$\Delta A_{2} B_{2} A_{3}$$, $$\Delta A_{3} B_{3} A_{4}$$, … are all equilateral triangles. If $$O A_{1} = 2$$, then the side length of $$\Delta A_{5} B_{5} A_{6}$$ is        .

Answer: 32


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, $$M$$ and $$N$$ are two points on $$\overset{⌢}{AB}$$ (distinct from $$A$$ and $$B$$), $$C$$ is a moving point on $$\overset{⌢}{MN}$$, the angle bisector of $$\angle ACB$$ intersects $$\bigodot O$$ at point $$D$$, and the angle bisector of $$\angle BAC$$ intersects $$CD$$ at point $$E$$. When point $$C$$ moves from point $$M$$ to point $$N$$, the ratio of the lengths of the paths traced by points $$C$$ and $$E$$ is           .

Answer: $$\sqrt{2}$$


Square $$ABCD$$ has side length 8, and points $$E$$, $$F$$ lie on sides $$AD$$, $$BC$$ respectively. Quadrilateral $$ABFE$$ is folded along $$EF$$, such that point $$A$$ lands at $$A'$$ and point $$B$$ lands at point $$B'$$, and $$A'B'$$ intersects $$BC$$ at $$G$$. The following conclusions: ① When $$A'$$ is the midpoint of $$CD$$, the side lengths of $$\triangle A'DE$$ are in the ratio $$3:4:5$$; ② Connect $$AA'$$, then $$AA' = EF$$; ③ When the side lengths of $$\triangle A'DE$$ are in the ratio $$3:4:5$$, $$A'$$ is the midpoint of $$CD$$; ④ As $$A'$$ moves along $$CD$$, the perimeter of $$\triangle A'CG$$ remains constant. Among these, the correct ones are           (write all correct conclusion numbers).

Answer: ①②④


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$A B = 3$$, $$A C = 4$$, $$\angle B A C = 90 \circ$$, $$D$$ and $$E$$ are moving points on sides $$A B$$ and $$A C$$ respectively, and $$B D = A E$$. Find the minimum value of $$C D + B E$$ .

Answer: $$\sqrt{58}$$


As shown in the figure, in parallelogram $$A B C D$$, points $$E$$ and $$F$$ are the midpoints of $$B C$$ and $$C D$$, respectively, $$\angle E A F = 60^\circ$$, $$A E = 3$$, $$A F = 6$$, then the length of $$A D$$ is      .

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


As shown in the figure, rhombus ABCD, $$AE \bot BC$$, point E is the foot of the perpendicular, point F is the midpoint of AE, connect BF and extend it to intersect AD at point G, connect CG, $$CE = 2, CG = 2\sqrt{11}$$, then the length of AF is          .

Answer: $$\sqrt{7}$$


As shown in the figure, point $$P$$ is a moving point on side $$BC$$ of rectangle $$ABCD$$. Triangle $$PAB$$ is folded along line $$AP$$, and point $$B$$ lands at position $$B^{'}$$. Given: $$AB = 6$$, $$BC = 4$$, find the length of $$BP$$ when point $$B^{'}$$ lies exactly on the axis of symmetry of the rectangle.

Answer: $$2 \sqrt{3}$$ or $$18 - 12 \sqrt{2}$$


In future studies, we will learn this theorem: in a right triangle, if one acute angle equals 30°, then the side opposite this angle equals half of the hypotenuse. That is, as shown, in Rt△ABC, ∠ACB=90°, if ∠ABC=30°, then $$A C = \frac{1}{2} A B$$. Problem: in Rt△ABC, ∠ACB=90°, ∠ABC=30°, AC=$$\sqrt{3}$$, point D is the midpoint of side BC, point E is a moving point on hypotenuse AB, connect DE, fold △BDE along line DE so that point B maps to point F. When line DF⊥AB, the length of AE is      .

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


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle B = \angle C = 90 \circ$$, point $$E$$ is a point on side $$B C$$, $$\triangle A D E$$ is an equilateral triangle, and $$\frac{A B}{C D} = \frac{n}{m}$$, then $$\frac{B E}{C E} =$$      .

Answer: $$\frac{2 m - n}{2 n - m}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 8$$, the diameter of semicircle O is $$B C$$. Point E starts from D and moves toward C at a speed of 1 unit per second. Point F starts from B and moves toward A at a speed of 2 units per second. When point F reaches point A, point E also stops moving. Let the time of motion be $$t$$ seconds.
(1) When $$E F$$ is tangent to semicircle O, $$t =$$
(2) Point M is the midpoint of $$E F$$, and point N is the circumcenter of $$\triangle M B C$$. The length of the path traced by point N is           .

Answer: 2     $$\frac{59}{36}$$


As shown in the figure, let AB be the diameter of semicircle O, and chords AC and BD intersect at point E. Point F lies on AC such that ∠DFE = ∠B. Given DF = 1 and cos ∠AED = √2/3, find the length of segment AB.

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


As shown in the figure, points E and F lie on sides AB and BC of rectangle ABCD, respectively. Connect EF. Fold triangle BEF along line EF to obtain triangle HEF. AB = 8, BC = 6, AE : EB = 3 : 1. Connect AH and HC. When point F moves along segment BC, the minimum area of quadrilateral AHCD is      .

Answer: $$32$$


As shown in the figure, the central angle of sector $$OAB$$ is $$\angle AOB = 45^\circ$$, and $$OB = 2$$. The sector $$OAB$$ is translated along ray $$OA$$ to obtain sector $$O'A'B'$$, where arc $$\overset{⌢}{AB}$$ intersects $$O'B'$$ at point C. If point C is the trisection point of arc $$\overset{⌢}{AB}$$ closer to point A, then the area of the shaded region is              .

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


As shown in the figure, given that the side length of square $$A B C D$$ is $$6$$, $$E$$ is a point on side $$A B$$ such that $$B E = 4$$, and $$F$$ is a point on side $$B C$$. Fold $$\triangle E B F$$ along $$E F$$ so that the image of point $$B$$ is $$B^{'}$$. Connect $$A C$$, $$D F$$, $$D E$$, and $$B^{'} D$$. The following conclusions are given: ① If $$E F \parallel A C$$, then $$D F = 2 \sqrt{10}$$; ② If $$E F \parallel A C$$, then $$B^{'} D = 2 \sqrt{2}$$; ③ The maximum area of $$\triangle D E F$$ is $$18$$; ④ The minimum value of $$B^{'} D$$ is $$2 \sqrt{10} - 4$$. Among these, the correct ones are     .

Answer: ①②③④


As shown in the figure, in rectangle $$A B C D$$, $$A D = 2 A B = 4$$. $$E$$ is the midpoint of $$B C$$, and $$F$$ is a moving point on segment $$A B$$ (excluding endpoints). Connect $$A E, D E, D F, E F$$. Let $$A E$$ and $$D F$$ intersect at point $$G$$, and let $$M$$ be the midpoint of $$D G$$. Connect $$M E$$ and $$M C$$. Among the following conclusions: ① $$M C$$ is the perpendicular bisector of $$D E$$; ② Points $$A, D, E, F$$ may be concyclic; ③ The minimum perimeter of $$\triangle M E F$$ is $$2 \sqrt{10}$$; ④ If $$F$$ is the midpoint of $$A B$$, then $$\triangle M E G$$ is an equilateral triangle. The correct ones are      . (Write all correct conclusion numbers)

Answer: ①③


As shown in the figure, point E is a moving point on side BC of square ABCD with side length 10. Connect DE, rotate segment DE counterclockwise 90° about point E to obtain segment EF, connect AF and DF. DF intersects AB at point G, and connect EG. When AF + DF is minimized, the length of segment EG is        .

Answer: $$\frac{25}{3}$$/$$8 \frac{1}{3}$$


As shown in the figure, in the equilateral triangle $$\Delta A B C$$, $$A B = 10$$, $$B D = 4$$, $$B E = 2$$. Point $$P$$ starts from point $$E$$ and moves along the direction of $$E A$$. Connect $$P D$$, and construct an equilateral triangle $$\Delta D P F$$ on the right side of $$P D$$ in the manner shown in the figure. When point $$P$$ moves from point $$E$$ to point $$A$$, the length of the path traced by point $$F$$ is  .

Answer: 8


As shown in the figure, the right triangle $$B E F$$ has vertex $$F$$ moving along the diagonal $$A C$$ of rectangle $$A B C D$$. Connect $$A E$$. $$\angle E B F = \angle A C D$$, $$A B = 6$$, $$B C = 8$$, then the minimum value of $$A E$$ is     .

Answer: $$\frac{72}{25}$$


As shown in the figure, △ABC is an isosceles triangle with AB = AC and ∠B = 30°. △ADE is a right triangle with ∠ADE = 90° and ∠E = 30°, and AD = AB. Rotate △ADE around point A so that AD and AE intersect BC at points F and G, respectively. When ∠AGB = 75°, $$\frac{F G}{D E} =$$     .

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


As shown in the figure, segment $$AD$$ intersects segment $$BC$$ at point $$E$$, and connect $$AB$$, $$CD$$. If $$\angle AEB = 60^{\circ}$$, $$AD = 2$$, $$BC = 3$$, then the minimum value of $$AB + CD$$ is        .

Answer: $$\sqrt{7}$$


As shown in the figure, given square $$A B C D$$, points P and Q move along $$B D$$ and $$C D$$ respectively. Connect $$A P$$, $$A Q$$, and $$P Q$$, where $$A Q$$ intersects $$B D$$ at point E, and $$\angle P A Q = 45^\circ$$. Reflect $$\triangle P Q E$$ over $$P Q$$ to obtain $$\triangle P Q E^{'}$$. When point $$E^{'}$$ lies on $$B C$$, $$B P = \sqrt{2}$$, then $$C Q =$$  , $$C E^{'} =$$      .

Answer: 2     $$\sqrt{5} - 1$$/$$- 1 + \sqrt{5}$$


As shown in the figure, it is given that $$BE$$ and $$CE$$ bisect $$\angle ABC$$ and $$\angle ACD$$ respectively, and $$\angle BEC = 30$$ degrees. Then $$\angle EAC =$$      degrees.

Answer: 60


As shown in the figure, in the right triangle $$\triangle$$ABC, $$\angle$$ACB = 90°, AB = 2AC, AC = $$\sqrt{2}$$, point E is a point on side AB. Fold $$\triangle$$BCE along CE to obtain $$\triangle$$B'CE. Draw AF parallel to BC, intersecting the angle bisector of $$\angle$$ABC at point F. Connect $$B'F$$. The minimum length of $$B'F$$ is

Answer: $$\sqrt{10} - \sqrt{6}$$


As shown in the figure, in $$\triangle A B C$$, two isosceles right triangles $$\triangle A D B$$ and $$\triangle A E C$$ are constructed on the same side with $$A B$$ and $$A C$$ as their hypotenuses, respectively. If point $$D$$ is the centroid of $$\triangle A E C$$, then $$t a n \angle B A C =$$      .

Answer: $$\frac{1}{2}$$/$$0 . 5$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points $$E$$, $$F$$ are moving points on sides $$B C$$, $$C D$$ respectively. Connect $$A E$$, $$A F$$, and fold the rectangle along $$A E$$, $$A F$$ such that the corresponding sides of $$A B$$, $$A D$$, namely $$A B^{'}$$, $$A D^{'}$$, lie on the same straight line. If point $$F$$ is the midpoint of $$C D$$, then $$A E =$$         .

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


As shown in the figure, $$\angle A B C$$ is a steel frame structure, with at most 5 equal-length steel bars that can be constructed inside the angle, satisfying $$B D = D E = E F = F G = G H = H I$$. Let $$\angle A B C = x$$, then the range of values for x is      .

Answer: $$15 \circ \leq x < 18 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B C A = 90 \circ$$, $$A B = 2 A C$$, point P is inside $$\triangle A B C$$, and $$P A = \sqrt{3}$$, $$P B = 5$$, $$P C = 2$$. Find the area of $$\triangle A B C$$   .

Answer: $$\frac{6 + 7 \sqrt{3}}{2}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, point D is the midpoint of $$A B$$, connect $$D C$$, and draw $$D E \bot D C$$ intersecting $$A C$$ at point E. If $$A B = 10$$, $$C E = 6$$, then the length of $$A E$$ is       .

Answer: $$\frac{7}{3}$$/$$2 \frac{1}{3}$$


As shown in the figure, in △ABC, ∠B = 30°, ∠ACB = 90°, AB = 2, D is on BC. Rotate segment AD counterclockwise 60° about point A to obtain AP. Then the minimum value of CP is     .

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


In $$\triangle ABC$$, $$\angle ACB = 90^\circ$$, $D$ is a point on $AB$, $AD = AC$, $E$ is a point on $AD$, through $E$ draw $EF \bot CD$, intersecting $CD$ at point $F$ and intersecting $BC$ at point $G$, $CG = BD$, $BC = 7$, $DE = 2$, then $AC = $      .

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


As shown in the figure, in parallelogram ABCD, E is the midpoint of CD, connect BE, F is the midpoint of BE, connect AF. If $$AB = 2$$, $$BC = 5$$, $$\angle BAD = 120^\circ$$, then the length of AF is        .

Answer: $$\frac{\sqrt{\text{19}}}{\text{2}}$$


As shown in the figure, in the equilateral triangle △ABC with side length 2, D is the midpoint of BC, and E is a point on side AC. The minimum value of BE + DE is     .

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


As shown in the figure, in the isosceles triangle $$\triangle ABC$$, $$AB = AC$$, $$\angle BAC = 45^\circ$$, $$AD \bot BC$$ at $$D$$. From point $$C$$, draw $$CE \bot AB$$ at $$E$$, intersecting $$AD$$ at $$H$$; draw $$EF \bot AC$$ at $$F$$, intersecting $$AD$$ at $$G$$; connect $$BH$$. The following conclusions: ① $$S_{\triangle AEH} : S_{\triangle ACH} = AE : AC$$; ② $$EG \parallel BH$$; ③ $$AH = 2CH$$; ④ $$\frac{1}{2} AD = EF$$; ⑤ $$AC = CE + GE$$. Among these, the correct ones are  .

Answer: ①②⑤


As shown in the figure, it is given that rectangle $$A B C D$$ has two sides $$A B = 6$$, $$A D = 8$$, point $$E$$ is the intersection of diagonals $$A C$$ and $$B D$$, and point $$P$$ is a moving point on side $$A D$$. Construct the reflection of point $$D$$ over line $$P E$$, denoted as $$D^{'}$$. When $$E D^{'}$$ is perpendicular to one side of the rectangle, the length of $$P D$$ is      .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C = 6$$, point P is a moving point inside $$\triangle A B C$$ such that the area of $$\triangle A C P$$ is 3, and point Q is a moving point on $$A B$$. Find the minimum value of $$P B + P Q$$            .

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


As shown in the figure, the area of $$\triangle A D C$$ is 4, $$A D$$ bisects $$\angle B A C$$, and $$A D \bot B D$$ at point $$D$$. What is the area of $$\triangle A B C$$?

Answer: 8


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A B = A C$$, draw $$C D \bot B C$$, connect $$D A$$, $$D B$$, draw $$A E \bot B D$$ through point $$A$$, meeting at point $$E$$, if $$\angle E A D = 2 \angle A D C$$, and the area of $$\triangle A D C$$ is 6, then the length of $$B C$$ is     .

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


As shown in the figure, in the equilateral triangle $$\triangle ABC$$ with side length 30, point M is a moving point on segment $$AB$$. The equilateral triangle $$\triangle ABC$$ is folded along a line passing through M, with the crease intersecting line $$AC$$ at point N, such that point A lands at point D on line $$BC$$, and $$BD : DC = 1 : 4$$. Let the crease be $$MN$$, then the value of $$AN$$ is      .

Answer: 21 or 65


As shown in the figure, in quadrilateral ABCD, ∠BAD = 90°, ∠ADC = 90°, ∠BCD = 60°, BC = CD. P is any point on the sides of quadrilateral ABCD. When AB = 4 and ∠APB = 30°, the length of BP is          .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = 4$$, $$A C = 3$$, points $$D$$ and $$E$$ lie on $$A B$$ and $$A C$$ respectively, with $$B D = C E$$; $$M$$ and $$N$$ are the midpoints of $$B E$$ and $$C D$$ respectively; line $$M N$$ intersects $$A B$$ and $$A C$$ at points $$P$$ and $$Q$$; if $$P D = 1$$, then $$Q E =$$      .

Answer: 2


As shown in the figure, rectangle $$A B C D$$, with side $$A B = 2$$, $$B C = 2 \sqrt{3}$$, fold the rectangle $$A B C D$$ so that point B lands on ray $$B D$$, and denote the corresponding point of B as $$B^{'}$$; the crease intersects sides $$A D$$ and $$B C$$ at points E and F respectively. When $$B^{'} D = 1$$, the length of $$A E$$ is       .

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


As shown in the figure, in square $$ABCD$$, points $$M$$ and $$N$$ are moving points on sides $$BC$$ and $$CD$$ respectively (excluding endpoints), and $$\angle MAN = 45^\circ$$. The following four conclusions: ① When $$MN = \sqrt{2} MC$$, then $$\angle BAM = 22.5^\circ$$; ② $$\angle AMN + \angle MNC = 90^\circ$$; ③ The perimeter of $$\triangle MNC$$ remains constant; ④ If $$DN = 2$$ and $$AB = 6$$, then the area of $$\triangle ABM$$ is 9. The correct conclusion(s) is/are           .

Answer: ①③④


As shown in the figure, in square $$ABCD$$, points $$E$$ and $$F$$ lie on $$AC$$ such that $$\frac{AE}{AC} = \frac{1}{3}$$ and $$\frac{CF}{AC} = \frac{1}{4}$$. Extend $$DE$$ to intersect $$AB$$ at point $$G$$, and extend $$DF$$ to intersect $$BC$$ at point $$H$$. Connect $$GH$$. The following conclusions: ① Point $$G$$ is the midpoint of $$AB$$, ② $$DF = GH$$, ③ $$\angle GDH = 45^\circ$$, ④ $$DE \cdot DG = DF \cdot DH$$. The correct conclusions are ______. (Write all correct conclusion numbers)

Answer: ①③④


As shown in the figure, in $$\triangle A B C$$, O is the intersection point of the perpendicular bisectors of the three sides of triangle $$A B C$$. If $$\angle A = 60^\circ$$, $$A B = 4$$, $$A C = 3$$, and segment $$O B$$ is connected, then the length of $$O B$$ is        .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$A C = 4$$, $$B C = 3$$, point $$D$$ is the midpoint of $$A C$$, and point $$E$$ is a moving point on side $$A B$$. Connect $$D E$$, and fold $$\triangle A D E$$ along the line $$D E$$ to obtain $$\triangle A^{'} D E$$. When $$A^{'} E \parallel B C$$, the length of $$A E$$ is      .

Answer: 1 or 4


As shown in the figure, the side length of square $$A B C D$$ is 4, and $$\triangle B E F$$ is an equilateral triangle, with point F located above side $$B C$$, and point E moving along ray $$B C$$. Connect $$C F$$, and let M be the midpoint of $$C F$$. Then the minimum length of segment $$A M$$ is      .

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


As shown in the figure, in square $$ABCD$$, $$AB = 10$$, point P is a moving point inside square $$ABCD$$. Connect $$AP$$, rotate $$AP$$ counterclockwise around point A by $$90^\circ$$ to obtain segment $$AQ$$, connect $$QD$$ and $$BP$$, extend $$BP$$ to intersect line $$QD$$ at point M. When point P is the midpoint of $$BM$$, the minimum value of segment $$PC$$ is      .

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


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, $$\triangle A B D$$ is an isosceles right triangle, $$\angle B A D = 90 \circ$$, $$A E \bot B D$$ at point $$E$$, connect $$C D$$ intersecting $$A E$$ and $$A B$$ at points $$F$$ and $$G$$ respectively, draw $$A H \bot C D$$ from point $$A$$ intersecting $$C D$$ and $$B D$$ at points $$P$$ and $$H$$ respectively, then the correct conclusion(s) is/are        .
①$$\angle B A C = 4 \angle A D C$$; ②$$D F = A H$$; ③$$B H = E F$$; ④$$\angle D A P = \angle C G B$$

Answer: ①②④


As shown in the figure, in Rt△ABC, ∠C=90°, AC=8, BC=6. Rotate △ABC about point C to obtain △A'B'C, where A' is the image of point A. Let P be the midpoint of A'B', and connect BP. During the rotation, the maximum length of segment BP is      .

Answer: 11


As shown in the figure, in rectangle $$A B C D$$, points $$E$$ and $$F$$ are on sides $$A B$$ and $$B C$$ respectively, $$B E = 4$$, $$C D = 8$$, $$\angle F E D = 30 \circ$$, $$\angle F D E = 45 \circ$$, then the length of $$B C$$ is      .

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


As shown in the figure, in parallelogram $$A B C D$$, $$A B = 2$$, $$A D = 4$$, $$\angle A = 60 \circ$$, $$E$$ is the midpoint of $$A D$$, $$F$$ is a moving point on $$E C$$, $$G$$ is the midpoint of $$B F$$, connect $$G D$$, then the minimum value of $$G D$$ is

Answer: $$2$$


As shown in the figure, points D and E are two points inside $$\triangle$$ABC, and DE$$/ /$$AB. Connect AD, BE, and CE. If AB=9$$\sqrt{2}$$, DE=2$$\sqrt{2}$$, BC=10, ∠ABC=75°, then the minimum value of AD+BE+CE is           .

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


As shown in the figure, in $$\triangle A B C$$, $$A C = B C$$, $$\angle A C B = 90 \circ$$, point D is the midpoint of AB, point E lies on AC, point F lies on BC, $$\angle E D F = 90 \circ$$, connect BE. If $$\angle C B E = 2 \angle E D A$$, $$C E = 6$$, then $$B E =$$          .

Answer: 10


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A D \bot B C$$ at point D, E is the midpoint of side $$A C$$, connect $$B E$$ intersecting $$A D$$ at F, fold $$\triangle A F E$$ along $$A C$$ to $$\triangle A G E$$, such that $$G E \parallel A D$$. The following conclusions: ① Quadrilateral $$A F E G$$ is a rhombus; ② $$2 A E^{2} = C D \cdot B C$$; ③ $$S_{\triangle A B F} = S_{\triangle C B F}$$; ④ Connect $$B G$$ intersecting $$A D$$ at H, then $$B H = H G$$. Among the above conclusions, the correct ones are     . (Fill in the correct serial numbers).

Answer: ①③④


As shown in the figure, point A is a point outside the equilateral triangle BCD. Connect AB and AD such that AB = AD. From point A, draw AE ∥ CD, intersecting BC and BD at points E and F, respectively. If 3BD = 5AE and EF = 6, then the length of segment AE is      .

Answer: 9


As shown in the figure, in sector $$A O B$$, $$\angle A O B = 90^{\circ}$$, radius $$O A = 6$$, point $$F$$ is located at $$\frac{1}{3}$$ of arc $$A B$$ closer to point $$A$$, points $$C$$ and $$D$$ lie on segments $$O A$$ and $$O B$$ respectively, $$C D = 6$$, $$E$$ is the midpoint of $$C D$$, connect $$E F$$ and $$B E$$. As $$C D$$ slides (with $$C D$$ length remaining constant), when $$E F$$ takes its minimum value, the perimeter of the shaded region is          .

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


As shown in the figure, in square $$A B C D$$, points $$E$$ and $$N$$ are the midpoints of $$B C$$ and $$A B$$, respectively. $$C N$$ intersects $$D E$$ at point $$G$$. Connect $$B G$$ and extend it to intersect $$C D$$ at point $$F$$. $$C N$$ intersects diagonal $$B D$$ at point $$H$$. The following conclusions are given: ① $$\angle C N B + \angle B E G = 180^\circ$$; ② $$S_{A B C D} = 6 S_{\triangle B N H}$$; ③ $$\frac{N G}{C D} = \frac{E G}{C F}$$; ④ $$N G + E G = \sqrt{2} B G$$. Among these, the correct conclusions are     . (Fill in all the serial numbers of the correct conclusions)

Answer: ①③④


As shown in the figure: In △ABC, AB = AC, ∠BAC = 90°, the vertex P of the right angle ∠EPF is the midpoint of side BC, and the two sides PE and PF intersect AB and AC at points E and F, respectively. The following four conclusions are given:

① AE = CF; ② EF = AP; ③ 2S_{quadrilateral AEPF} = S_{△ABC}; ④ When ∠EPF rotates around vertex P within △ABC (point E does not coincide with A or B), BE + CF = EF. Among the above conclusions, the always correct ones are          .

Answer: ①③


As shown in the figure, in equilateral triangle △ABC, BF is the median on side AC, and point D is a moving point on BF. Connect AD, and construct equilateral triangle △ADE on the right side of AD. Connect EF. When the perimeter of △AEF is minimized, the measure of ∠CFE is     .

Answer: $$90^{\circ}$$


As shown in the figure, in $$\triangle ABC$$, $$AB = AC$$, $$\angle B = 40^\circ$$, point $$D$$ is a moving point on $$BC$$, and $$\triangle ABD$$ is folded along $$AD$$ to obtain $$\triangle ADE$$. When the overlapping part of $$\triangle ADE$$ and $$\triangle ABC$$ is a right triangle, the measure of $$\angle BAD$$ is                 .

Answer: $$25 \circ$$ or $$50 \circ$$ or $$75 \circ$$


As shown in the figure, quadrilateral $$A B C D$$ is a parallelogram, point $$E$$ is a point on side $$C D$$, and $$B C = E C$$. $$C F \bot B E$$ intersects $$A B$$ at point $$F$$ and intersects $$B E$$ at point $$G$$. Point $$P$$ is a point on the extension of $$E B$$. The following conclusions are given: ① $$B E$$ bisects $$\angle C B F$$; ② $$C F$$ bisects $$\angle D C B$$; ③ $$B C = P B$$; ④ $$P F = P C$$. Among these, the correct ones are            . (Fill in the serial numbers)

Answer: ①②④


As shown in the figure, in square $$A B C D$$, points $$E$$, $$F$$ lie on sides $$B C$$, $$C D$$ respectively, and $$\angle E A F = 45^\circ$$. $$A E$$ intersects $$B D$$ at point $$M$$, $$A F$$ intersects $$B D$$ at point $$N$$. The extension of $$F M$$ intersects the extension of $$C B$$ at point $$P$$, and $$B P = D F$$. Connect $$E F$$.

(1) $$\angle A M F =$$        .
(2) If $$D F = 2$$, $$E F = 5$$, then $$t a n \angle C F E =$$        .

Answer: $$90 \circ$$     $$\frac{3}{4}$$


As shown in the figure, the diameter AB of semicircle O is 15, chord BC is 9, and chord BD bisects ∠ABC. Then the length of BD is          .

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


Given $$\triangle A B C$$, $$\angle C = 90 \circ$$, point $$E$$ is the midpoint of $$A B$$, $$E F \bot A F$$, $$\angle B A C = \angle C A F$$, if $$B C = 2 \sqrt{3}$$, $$A F = 1$$, then $$E F =$$          .

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


As shown in the figure, in the isosceles right triangle $$\text{Rt} \triangle A B C$$, $$A B = B C$$, $$\angle A B C = 90 \circ$$, points $$D$$ and $$E$$ are moving points on sides $$A C$$ and $$B C$$ respectively, satisfying $$C E = \sqrt{2} A D$$. Find the minimum value of $$\frac{D E}{D B}$$.

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


As shown in the figure, points $$D$$ and $$E$$ are on sides $$BC$$ and $$AC$$ of $$\triangle ABC$$, respectively. Connect $$DE$$. Fold $$\triangle CDE$$ along $$DE$$ so that point $$C$$ lands at point $$F$$. Connect $$BF$$ and $$CF$$. If $$EF \parallel AB$$, $$BF \bot CF$$, $$AB = DF = 3\sqrt{5}$$, $$FC = 12$$, $$DE = 5$$, then the length of $$AE$$ is      .

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


In $$\triangle ABC$$, $$AB = 4$$, $$\angle ABC = 60^\circ$$, $$\angle ACB = 45^\circ$$, D is the midpoint of $$BC$$, line $$l$$ passes through point D, from B draw $$BF \bot l$$ to F, from A draw $$AE \bot l$$ to E, find the maximum value of $$AE + BF$$ is      .

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


As shown in the figure, it is given that quadrilateral $$A B C D$$ is a square, $$A B = 3$$, and $$E$$ is a point on diagonal $$A C$$. Connect $$D E$$, and from point $$E$$ draw $$E F \bot D E$$, intersecting the extension of $$B C$$ at point $$F$$. Construct rectangle $$D E F G$$ with $$D E$$ and $$E F$$ as adjacent sides. Connect $$C G$$, and given that $$\tan \angle C G F = \frac{1}{3}$$, then $$A E =$$         .

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


As shown in the figure, the square $$ABCD$$ with side length 4 is rotated counterclockwise about point $$A$$ to obtain the square $$AB^{'}C^{'}D^{'}$$. Connect $$BB^{'}$$ and $$BC^{'}$$. During the entire rotation process from 0° to 180°, when $$BB^{'} = BC^{'}$$, the area of $$\triangle BB^{'}C^{'}$$ is            .

Answer: $$8 + 4 \sqrt{3}$$ or $$8 - 4 \sqrt{3}$$


As shown in the figure, $$BC$$ is the diameter of $$\bigodot O$$ with radius 5, and A, D are two points on the circle. $$AE \bot BC$$ at point E, $$DF \bot BC$$ at point F, $$AE = 3$$, $$DF = 4$$. Point P is a moving point on $$BC$$. $$AP$$ intersects $$DE$$ at point Q. When $$PA + PD$$ is minimized, the length of $$AQ$$ is          .

Answer: $$\frac{21}{11} \sqrt{2}$$


As shown in the figure, in rectangle ABCD, AB = 3, AD = 4, point E is the midpoint of AD, point P is a moving point on BE, point Q is the midpoint of PC, connect AQ, then the minimum length of AQ is      .

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


As shown in the figure, point F is a point inside rectangle $$A B C D$$, and point E lies on side $$B C$$. Connect $$A F$$ and $$E F$$. Rotate segment $$A F$$ clockwise around point $$A$$ by $$90^\circ$$ to obtain $$A P$$, and connect $$P E$$. If $$A B = 8$$, $$B C = 6$$, $$B E = \frac{1}{2} C E$$, $$E F = 4$$, then the minimum value of segment $$P E$$ is          .

Answer: $$2 \sqrt{34} - 4$$


As shown in the figure, in square ABCD, E is a moving point on CD. Fold the square along BE such that point C lands at point F. Connect CF and extend it to intersect AD at point G. Extend BF to intersect side AD at point H. If $$\frac{C E}{B C}$$=$$\frac{\text{2}}{\text{3}}$$, then the value of $$\frac{G H}{D H}$$ is        .

Answer: $$\frac{\text{1}}{\text{7}}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle A = 30 \circ$$, $$A C = 6$$, D is a moving point on AB, and an isosceles right triangle $$\triangle D C E$$ is constructed with DC as the hypotenuse, such that $$\angle C E D = 90 \circ$$, and points E and A lie on opposite sides of CD. Connect BE; the minimum value of BE is      .

Answer: $$\frac{\sqrt{6}}{2}$$/$$\frac{1}{2} \sqrt{6}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 60 \circ$$, $$\angle A B C < 60 \circ$$, the three angle bisectors $$A D$$, $$B E$$, $$C F$$ intersect at point O, $$O H \bot B C$$ at point H. The following conclusions: ① $$\angle B O C = 120 \circ$$, ② $$\angle D O H = \angle O C B - \angle O B C$$, ③ $$O D$$ bisects $$\angle B O C$$, ④ $$O E = O F$$, among which the correct conclusion numbers are               .

Answer: ①②④


Given that the side length of square $$A B C D$$ is 4, point $$P$$ is a moving point on segment $$A D$$ (not coinciding with point $$A$$), and point $$E$$ is the reflection of point $$A$$ across line $$B P$$. Connect $$P E$$, $$B E$$, $$C E$$, and $$D E$$. When $$\triangle C D E$$ is an isosceles triangle with $$C E$$ as one of its equal sides, the value of $$A P$$ is      .

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


As shown in the figure, there is a square $$GICJ$$ at the bottom-right corner of the square $$ABCD$$. Construct a square $$EFGH$$ with vertex $$G$$ extending to the left, such that points $$E$$ and $$F$$ lie on sides $$AB$$ and $$BI$$, respectively. When points $$A$$, $$H$$, and $$J$$ are collinear, the value of $$\frac{GF}{GI}$$ is       .

Answer: $$\sqrt{3}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, construct a square $$A D E B$$ downward using the hypotenuse $$A B$$ as a side. Draw $$E F \parallel B C$$ intersecting $$A C$$ at point F. Draw $$C G \parallel B E$$ intersecting $$E F$$ at point G. Connect $$D G$$. If $$A F = 3$$ and $$D E = 15$$, then the quantitative relationship between segments $$A D$$ and $$C G$$ is                 ; the area of quadrilateral $$C G E B$$ is        .

Answer: $$A D \parallel C G$$ and $$A D = C G$$     $$81$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 30 \circ$$, $$A C = 4 \sqrt{3}$$, $$A B = 8$$, point $$D$$ is inside $$\triangle A B C$$, connect $$D A$$, $$D B$$, $$D C$$, then the minimum value of $$D C + D B + \sqrt{3} A D$$ is      .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2 B C$$, point $$M$$ is the midpoint of side $$C D$$, and points $$E, F$$ are points on sides $$A B$$ and $$B C$$ respectively, such that $$A F \bot M E$$ at point $$G$$. If $$E B = 2$$ and $$B F = 1$$, then the area of quadrilateral $$B F G E$$ is      .

Answer: $$\frac{85}{52}$$


As shown in the figure, in parallelogram ABCD, AB = 6, BC = 4, ∠A = 120°, E is the midpoint of AB, and point F lies on the sides of parallelogram ABCD. If triangle AEF is an isosceles triangle, then the length of EF is     .

Answer: 3$$\sqrt{3}$$ or 3 or $$\frac{\sqrt{57}}{2}$$


As shown in the figure, $$AB$$ is the diameter of a semicircle, $$AB = 10$$, the distance from point $$O$$ to chord $$AC$$ is $$4$$, point $$P$$ starts from $$B$$ and moves toward point $$A$$ along $$BA$$ at a speed of $$1$$ unit per second, connecting $$CP$$, after      seconds, $$\Delta APC$$ is an isosceles triangle.

Answer: $$\frac{14}{5}$$ or $$4$$ or $$5$$


As shown in the figure, in $$\Delta A B C$$, $$\angle A C B = 45 \circ$$, $$A B = 4$$, points $$E$$ and $$F$$ lie on sides $$B C$$ and $$A B$$ respectively, point $$E$$ is the midpoint of side $$B C$$, $$A B = 3 A F$$, connect $$A E$$ and $$C F$$ intersecting at point $$P$$, then the maximum area of $$\Delta A B P$$ is   .

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


As shown in the figure, OP bisects ∠AOB, ∠AOP = 15°, PC ∥ OA, PC = 4, and point D is a moving point on ray OA. Then the minimum value of PD is     .

Answer: 2