As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle C A B = 90 \circ$$, D and E are points on sides $$A C$$ and $$A B$$ respectively, and $$A D = A E$$. Connect $$B D$$ and $$C E$$. Let $$A F \bot C E$$ intersect at point G and intersect $$B C$$ at point F. Let $$F P \bot B D$$ intersect $$C E$$ at point H and intersect $$A B$$ at point P. Connect $$A H$$. If $$A H$$ bisects $$\angle B A F$$, then the following conclusions: ① $$A P = B E$$; ② $$\angle H P E = \angle H E P$$; ③ $$A P + A D = 2 A G$$; ④ $$C H = A F + F H$$; ⑤ $$\angle D B C = 15 \circ$$. The correct conclusions are numbered

Answer: ①②③④⑤


Definition: In a triangle, if the measure of one interior angle is three times the measure of another interior angle, then such a triangle is called a "beautiful triangle." For example: a triangle with interior angles of $$100 \circ$$, $$60 \circ$$, and $$20 \circ$$ is a "beautiful triangle." As shown in the figure, point $$D$$ lies on side $$AB$$ of $$\triangle ABC$$, connect $$DC$$, with $$\angle BDC > 90^\circ$$. Construct the angle bisector of $$\angle ADC$$, intersecting $$AC$$ at point $$E$$. On $$DC$$, take a point $$F$$ such that $$\angle EFC + \angle BDC = 180^\circ$$ and $$\angle DEF = \angle B$$. If $$\triangle BCD$$ is a "beautiful triangle", then $$\angle B$$ equals        .

Answer: $$36 \circ$$


As shown in the figure, in $$\triangle A B C$$ and $$\triangle D E F$$, $$\angle A C B = \angle D F E = 90 \circ$$, $$\angle A = \angle E D F = 60 \circ$$, and $$E D$$ moves along the line $$A B$$. If $$A C = E D = 2$$, then the minimum value of $$C E + C F$$ is      .

Answer: $$\sqrt{21}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 3$$, $$B C = 4$$, point $$F$$ lies on $$B C$$ with $$B F = 1$$, and point $$E$$ moves along segment $$A B$$ (not coinciding with $$A$$ or $$B$$). Segment $$E F$$ is rotated clockwise around point $$F$$ by $$30^{\circ}$$ to obtain $$F G$$, and $$C G$$ is connected.

(1) When $$B E = \sqrt{3}$$, then $$C G =$$            ; (2) During the motion of $$E$$, the minimum value of $$C G$$ is          .

Answer: $$\sqrt{13}$$     $$1 + \frac{3 \sqrt{3}}{2}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = 2$$, $$B D \bot A C$$, point E lies on line $$B D$$, $$B E = A C$$, $$\angle C E A = 90 \circ$$, then the minimum value of $$B C$$ is        .

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


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = 2$$, $$B C = 3$$, $$\angle B = 60^{\circ}$$, and $$P$$ is a moving point on side $$B C$$ ($$B P > 1$$). Fold $$\triangle A B P$$ along $$A P$$ to obtain $$\triangle A B^{'} P$$. The ray $$P B^{'}$$ intersects the ray $$A D$$ at point $$E$$. The following statements are given: (1) When $$A B^{'} \bot A B$$, $$B^{'} A = B^{'} E$$; (2) When point $$B^{'}$$ lies on $$A D$$, quadrilateral $$A B P B^{'}$$ is a rhombus; (3) As point $$P$$ moves, the minimum value of segment $$A E$$ is 2; (4) Connect $$B B^{'}$$, then the area of quadrilateral $$A B P B^{'}$$ is always equal to $$\frac{1}{2} A P \cdot B B^{'}$$. The correct statement(s) are     .

Answer: (1)(2)(4)


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle B = 90 \circ$$, $$\angle B C D = 45 \circ$$. Connect $$A C$$ and from point D draw a perpendicular to diagonal $$A C$$, intersecting $$B C$$ at point E and $$A C$$ at point F. If $$A B = B E$$, $$\angle D A C = 2 \angle A C B$$, $$C D = 3 \sqrt{2}$$, then the length of $$A C$$ is      .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 3$$, point $$M$$ is the midpoint of $$A B$$, and point $$N$$ is a moving point on side $$C D$$ (not coinciding with either endpoint). If quadrilateral $$A M N D$$ is folded along line $$M N$$ to obtain quadrilateral $$E F M N$$ (point $$E$$ and point $$F$$ correspond to point $$D$$ and point $$A$$, respectively), connect $$E C$$ and $$F C$$. When $$E C \bot F C$$, the perimeter of $$\triangle E C F$$ is      .

Answer: $$\frac{15 + 9 \sqrt{5}}{5}$$


As shown in the figure, in rectangle $$A B C D$$, point $$H$$ is the midpoint of $$C D$$, and lines $$B D$$ and $$A H$$ intersect at point $$G$$. From point $$A$$, draw a perpendicular to $$B D$$, intersecting $$B D$$ at point $$E$$, and extend $$A E$$ to intersect $$B C$$ at point $$F$$. If $$A B = B G$$, take a point $$P$$ on $$B D$$ (not coinciding with $$B$$ or $$D$$) such that line $$C P$$ passes through the midpoint of one side of quadrilateral $$D E F H$$. Find the value of $$B P : B D$$       .

Answer: $$\frac{13}{18}$$ or $$\frac{5}{7}$$ or $$\frac{10}{27}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle B = 90 \circ$$, $$A B = 4$$, $$B C = 3$$, first fold $$\triangle A B C$$ along $$A C$$ to $$\triangle A B^{'} C$$, then fold $$\triangle A B^{'} C$$ along $$A B^{'}$$ to $$\triangle A B^{'} C^{'}$$, draw $$C D \parallel A B$$ through point $$C$$ intersecting $$A C^{'}$$ at point $$D$$, then the length of $$C D$$ is          .

Answer: $$\frac{200}{39}$$


As shown in the figure, in parallelogram $$\square ABCD$$, $$AB = 2$$, $$BC = 4$$, $$\angle B = 30^\circ$$, point $$P$$ is a moving point on side $$BC$$, connect $$AP$$, rotate segment $$AP$$ clockwise about point $$P$$ by $$90^\circ$$; when the image point $$E$$ of point $$A$$ lies exactly on the line containing one of the sides of $$\square ABCD$$, the length of segment $$AP$$ is      .

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


As shown in the figure, the square $$ABCD$$ has side length 6, and points $$E$$ and $$F$$ are on sides $$BC$$ and $$CD$$ respectively, with $$BE = CF = 2$$. Connect $$AE$$ and $$AF$$. The perpendicular bisector of $$AE$$ intersects $$AB$$, $$AE$$, $$AF$$, and $$CD$$ at points $$G$$, $$M$$, $$N$$, and $$H$$, respectively. Find the length of $$MN$$      .

Answer: $$\frac{7}{9} \sqrt{10}$$


As shown in the figure, the radius of $$\bigodot O$$ is $$2 \sqrt{3}$$, $$AB$$ is the diameter of $$\bigodot O$$, a line $$DE \bot OB$$ is drawn through the midpoint $$C$$ of radius $$OB$$, intersecting $$\bigodot O$$ at points $$D$$ and $$E$$, point $$F$$ lies on $$\bigodot O$$, connect $$FD$$, draw $$HA \bot AB$$ from point $$A$$ intersecting the extension of $$FD$$ at $$H$$, if $$DH = 3 \sqrt{7}$$, then the length of segment $$AH$$ is        , and the length of segment $$EF$$ is           .

Answer: 9     $$\frac{12 \sqrt{7}}{7}$$


As shown in the figure, it is given that $$AB$$ is the diameter of $$\bigodot O$$, $$BC$$ and $$CD$$ are tangent lines to $$\bigodot O$$, with points of tangency at $$B$$ and $$D$$, respectively. Point $$E$$ is a moving point on $$AB$$. Connect $$CE$$ and $$DE$$. If $$AB = 2\sqrt{5}$$ and $$BC = 2$$, then the minimum value of $$CE + DE$$ is          .

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


As shown in the figure, the diameter of semicircle O is $$MN$$, point A lies on radius $$OM$$, B is the midpoint of arc $$\overset{⌢}{MN}$$, and point C lies on arc $$BN$$. A rectangle $$ABCD$$ is constructed with $$AB$$ and $$BC$$ as adjacent sides. Side $$CD$$ intersects $$MN$$ at point E. Connect $$DO$$ and extend $$DO$$ to intersect $$AB$$ at point P. If $$BP = 2AP$$, then the value of $$\frac{BC}{AB}$$ is                      .

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


As shown in the figure, in $$\text{Rt} \triangle A C B$$, $$\angle A C B = 90 \circ, A C = B C = 6, C D = 2, C H \bot B D$$ at H, point O is the midpoint of $$A B$$, connect $$O H$$, then $$O H =$$      .

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


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle with side length 2. Fold $$\triangle A B C$$ along the line $$A C$$ to obtain $$\triangle A B^{'} C$$, then translate $$\triangle A B^{'} C$$ along the line $$A C$$ to obtain $$\triangle A^{'} B^{' '} C^{'}$$. The minimum perimeter of $$\triangle B B^{' '} C^{'}$$ is       .

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


As shown in the figure, the side length of square $$A B C D$$ is 4. Take the midpoint $$E$$ of side $$A B$$, connect $$C E$$, draw $$B F \bot C E$$ through point $$B$$, intersecting at point $$F$$, connect $$D F$$, draw $$A H \bot D F$$ through point $$A$$, intersecting at point $$H$$, intersecting $$C E$$ at point $$M$$, and intersecting $$B C$$ at point $$N$$. Then (1) $$E F =$$                 ; (2) $$M N =$$                 .

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


As shown in the figure: In quadrilateral $$A B C D$$, $$\angle B A C = 90 \circ$$, $$\angle A D C =\text{45} \circ$$, point $$C$$ moves along segment $$D G$$, and $$\angle A B D = 2 \angle B D C$$, $$B D$$ intersects $$A C$$ at point $$E$$. If $$B D = 9$$, $$C B = 3 \sqrt{5}$$, $$A G$$ intersects $$D B$$ at point $$F$$, in triangle $$A D G$$, $$t a n \angle D A F = 1$$, then the sine of $$\angle B E A$$ is        .

Answer: $$\frac{3}{5}$$ or $$\frac{12}{13}$$


As shown in the figure, the rhombus $$A B C D$$ has side length $$6$$, $$\angle A B C = 60^\circ$$, and segment $$A C$$ is connected. Points $$E$$ and $$F$$ are moving points on segments $$A B$$ and $$A C$$ respectively (not coinciding with endpoints), and $$B E = A F$$. $$B F$$ intersects $$C E$$ at point $$P$$. Extend $$B F$$ to intersect side $$A D$$ (or side $$C D$$) at point $$G$$. The following conclusions are given: ① $$\triangle A B F \sim \triangle B C E$$; ② When $$B E = 4$$, $$B E : C G = 4 : 3$$; ③ Connect $$E F$$, the minimum area of quadrilateral $$B C F E$$ is $$6 \sqrt{3}$$; ④ When $$\angle P D C$$ is maximized, the length of segment $$D P$$ is $$6 \sqrt{2}$$. All correct conclusions are        .

Answer: ①②④


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 60 \circ$$, $$A B = 5$$, $$A C = 8$$, P is a point inside $$\triangle A B C$$, $$\angle B P C = 120 \circ$$, connect $$A P$$, then the minimum length of $$A P$$ is           .

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


As shown in the figure, given equilateral triangle $$\triangle A B C$$, $$A C = 4$$, construct square $$A C D E$$ with side $$A C$$ (points $$A$$, $$C$$, $$D$$, $$E$$ arranged counterclockwise), and let the extensions of $$B C$$ and $$E D$$ intersect at point $$F$$. Point $$P$$ starts from point $$B$$ and moves along $$B F$$ toward point $$F$$, stopping at $$F$$. Point $$Q$$ moves along segment $$A C$$ and $$C D$$, always satisfying that $$P Q$$ is perpendicular to a side of the square. Connect $$E Q$$, $$E P$$, and $$P Q$$. When $$\frac{E Q}{P Q} = \frac{5}{\sqrt{3}}$$, the area of $$\triangle E P Q$$ is                     .

Answer: $$\frac{3 \sqrt{3}}{2}$$ or $$\frac{17 \sqrt{123} - 99 \sqrt{3}}{16}$$ or $$\frac{4 \sqrt{141} - 12}{11}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$\angle B A D = 120^\circ$$, $$A B = 2$$, $$B C = 3$$, point E is a moving point on side $$B C$$, connect $$A E$$, rotate $$A E$$ clockwise around point E by $$60^\circ$$ to obtain $$F E$$, connect $$C F$$ and $$D F$$, then the minimum value of $$C F + D F$$ is       .

Answer: $$\sqrt{7}$$


As shown in the figure, in $$\triangle A B C$$, $$D$$ is the point dividing $$A B$$ into four equal parts closer to point $$B$$, connect $$C D$$, and take the midpoint $$E$$ of $$C D$$, connect $$B E$$. If $$\angle A C D = 30 \circ$$, $$\angle B E D = 60 \circ$$, $$C D = 2$$, then the length of $$A B$$ is      .

Answer: $$\frac{2 \sqrt{31}}{3}$$


As shown in the figure, $$\triangle A B D$$ and $$\triangle C B D$$ are both equilateral triangles. Draw a ray inside $$\angle A B C$$, and construct point $$E$$, the reflection of point $$C$$ over line $$B M$$. Connect $$A E$$ and extend it to intersect $$B M$$ at point $$F$$. Connect $$C E$$ and $$C F$$. If $$A E = 5$$ and $$E F = 2$$, then the length of $$B F$$ is      .

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


As shown in the figure, $$\triangle A B C$$ is an inscribed triangle of $$\bigodot O$$, $$A C > B C$$, $$\angle A C B = 45 \circ$$, and $$\triangle A B C$$ is rotated counterclockwise about point A to obtain $$\triangle A D E$$ (with points B and C corresponding to points D and E, respectively). When $$A D$$ is tangent to $$\bigodot O$$, the line containing $$D E$$ is also tangent to $$\bigodot O$$. If the radius of $$\bigodot O$$ is 3, then the length of $$B C$$ is      .

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


As shown in the figure, in $$\triangle A B F$$, C and E are points on sides $$B F$$ and $$A F$$ respectively. Connect $$A C$$ and $$B E$$, intersecting at point D. If $$A B = A C = A E = 12$$, $$A D = 9$$, $$\angle B A C = 60^\circ$$, then the length of segment $$E F$$ is          .

Answer: 7.5/$$\frac{15}{2}$$


Given in $$\triangle ABO$$, $$\angle B = 90^\circ$$, $$\angle AOB = 30^\circ$$, rotate $$\triangle ABO$$ around point $$O$$ by $$120^\circ$$ and $$240^\circ$$ to obtain $$\triangle DCO$$ and $$\triangle FEO$$, respectively. Connect $$AD$$, $$AF$$, $$FD$$, intersecting $$OC$$, $$OB$$, $$OE$$ at points $$P$$, $$Q$$, $$R$$ respectively. If the area of $$\triangle ADF$$ is $$10$$, then the area of $$\triangle PQR$$ is     .

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


As shown in the figure, in the right triangle $$\triangle ABC$$, $$AB = BC$$, point $$D$$ is a moving point on side $$AC$$, and an isosceles right triangle $$\triangle DBE$$ is constructed with $$BD$$ as one leg. $$DE$$ intersects $$BC$$ at point $$F$$, and $$CE$$ is connected. From point $$B$$, draw $$BQ \bot DE$$, intersecting at point $$P$$ and intersecting $$CD$$ at point $$Q$$. Among the following conclusions, the correct ones are ______.

① $$\triangle ABD \sim \triangle CBE$$;
② $$AD^{2} + CQ^{2} = DQ^{2}$$;
③ When $$AD : DC = 1 : 2$$, $$S_{\triangle BEC} + S_{\triangle DCE} = S_{\triangle DBE}$$;
④ When $$CD = BC$$, $$BD : EF = \sqrt{2} + 1$$.

Answer: ①②④


As shown in the figure, in rectangle $$A B C D$$, $$D E$$ bisects $$\angle A D C$$, intersecting $$B C$$ at point E, $$E F \bot A E$$, intersecting $$C D$$ at point F. Construct rectangle $$A E F G$$ with sides $$A E$$ and $$E F$$, and $$F G$$ intersects $$D A$$ at point H. If $$C E = 3$$, $$A H = 4$$, then $$A E =$$                .

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


As shown in the figure, $$\triangle A B C$$ is inscribed in $$\bigodot O$$, $$A B$$ is the diameter of $$\bigodot O$$, chord $$C D = C B$$, $$C E \bot A B$$ at point $$E$$, connect $$B D$$ intersecting $$C E$$ at point $$F$$, intersecting $$A C$$ at $$G$$. If $$C D = 4 \sqrt{5}$$, $$A C = 8 \sqrt{5}$$, which of the following statements are correct?
① $$\angle A C B = 90 \circ$$; ② $$C F = B F$$; ③ The area of $$\triangle A B G$$ is $$100$$; ④ $$B D = 16$$.

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


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 is a point on the extension of $$A E$$, and $$\angle A C F = \angle C A F$$, the extensions of the line segments intersect at point G. If $$A B = \sqrt{5}$$, $$A D = 4$$, $$\text{tan} \angle A B C = 2$$, then the length of $$B G$$ is          .

Answer: $$\frac{20 \sqrt{5}}{19}$$/$$\frac{20}{19} \sqrt{5}$$


As shown in the figure, in $$\triangle A B C$$, the altitude $$A D$$ bisects side $$B C$$ exactly, $$\angle B = 30^\circ$$, point $$P$$ is a point on the extension of $$B A$$, and point $$O$$ is a point on segment $$A D$$ such that $$O P = O C$$. The following conclusions: $$\textcircled{1} A B = A C$$, $$\textcircled{2} \triangle A O P \cong \triangle A O C$$, $$\textcircled{3} \angle A P O + \angle D C O = 30^\circ$$, $$\textcircled{4} \triangle O P C$$ is an equilateral triangle. Among these, the correct ones are ______. (Fill in the serial numbers)

Answer: $$\textcircled{1}\textcircled{3}\textcircled{4}$$


As shown in the figure, given that the side length of square $$A B C D$$ is 8, point $$E$$ is a moving point on side $$A D$$, connect $$E C$$, rotate $$E C$$ counterclockwise by $$90^{\circ}$$ about point $$E$$ to obtain $$E F$$, connect $$D F$$ and $$C F$$, then the minimum value of $$D F + C F$$ is        .

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


As shown in the figure, $$AB$$ is the diameter of $$\bigodot O$$, and $$AB = 2\sqrt{2}$$. Point $$C$$ lies on the semicircle, and $$OC \bot AB$$, with the foot of the perpendicular at point $$O$$. Point $$P$$ is any point on the semicircle. From point $$P$$, draw $$PE \bot OC$$, intersecting at point $$E$$. Let the incenter of $$\triangle OPE$$ be $$M$$. Connect $$OM$$ and $$PM$$. As point $$P$$ moves along the semicircle from point $$B$$ to point $$A$$, the length of the path traced by the incenter $$M$$ is      .

Answer: $$\pi$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 5$$, point $$E$$ lies on side $$A B$$ such that $$B E = 2$$, and point $$P$$ is a moving point inside the rectangle satisfying $$\angle P A B = \angle P B C$$. Connect $$P E$$, rotate $$P E$$ counterclockwise by $$90^\circ$$ about point $$P$$ and extend it to point $$P F$$ such that $$P F = 2 P E$$. Connect $$C F$$. Find the minimum value of $$C F$$          .

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


As shown in the figure, in the equilateral triangle $$\triangle ABC$$ with side length 3, point D is a moving point on side $$BC$$, satisfying $$CE \bot AD$$, with foot at point E. Point F is the symmetric point of point B with respect to point E. Point G is the trisection point on $$AB$$ closer to point A. Connect $$FG$$. When $$\angle AGF$$ is maximized, $$FG =$$            .

Answer: 2


As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$B D$$ and $$A C$$ intersect at point $$O$$. Point $$D$$ is rotated clockwise around point $$A$$ by $$60^\circ$$ to obtain point $$D^{'}$$. Connect $$O D^{'}$$ and $$C D^{'}$$. When the length of segment $$O D^{'}$$ is minimized, the length of $$C D^{'}$$ is $$\sqrt{3} - 1$$. Find the side length of the rhombus.

Answer: $$\sqrt{2}$$


As shown in the figure, in square $$A B C D$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$, $$E$$ and $$F$$ are points on $$C D$$ and $$B C$$ respectively, $$\angle E A F = 45^\circ$$, $$A E$$ and $$A F$$ intersect $$B D$$ at $$H$$ and $$G$$ respectively, $$A G \bot A I$$, $$D I \bot B D$$. Which of the following conclusions is correct? (Fill in the serial number).

① $$B F + D E = E F$$, ② $$A H^{2} + A G^{2} = O G^{2} + O H^{2} + A D^{2}$$, ③ $$B G^{2} + D H^{2} = G H^{2}$$, ④ $$G H \cdot B D = \frac{1}{2} A H \cdot G I$$, ⑤ $$B G^{2} + D G^{2} = B D^{2}$$

Answer: ①②③


As shown in the figure, in $$\triangle B C P$$, $$\angle B P C = 90 \circ$$, A lies on side $$P C$$, $$\angle B A C = 120 \circ$$, D is the midpoint of $$B C$$, $$A D = 4 \sqrt{3}$$, then the maximum value of segment $$P D$$ is       .

Answer: 12


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, point $$D$$ is a moving point on side $$A C$$, and point $$E$$ is a point on side $$B C$$ such that $$A D = C E$$. Connect $$A E$$ and $$B D$$. When the length of segment $$C F$$ is minimized, the value of $$\frac{S_{\triangle A B F}}{S_{\triangle A B C}}$$ is      .

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


As shown in the figure, A, F, B, C are four points on circle O, quadrilateral $$O A B C$$ is a parallelogram, $$\angle F A B = 15^\circ$$, connect $$O F$$ intersecting $$A B$$ at point E, draw the tangent to the circle at point C intersecting the extension of $$A B$$ at point D and the extension of $$A F$$ at point H. If $$O A = 3$$, then $$C H =$$     .

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


As shown in the figure, $$AD$$ is the altitude from vertex $$A$$ to side $$BC$$ in equilateral $$\triangle ABC$$, and point $$E$$ is a moving point on $$AD$$ (point $$E$$ does not coincide with point $$A$$). Connect $$CE$$, and construct an equilateral $$\triangle CEF$$ with $$CE$$ as a side. If $$AB = 6$$, then the minimum value of the length of segment $$DF + CF$$ is      .

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


As shown in the figure, in the rhombus $$A B C D$$, draw $$D E \bot A D$$ intersecting diagonal $$A C$$ at point $$E$$, and connect $$B E$$. Point $$P$$ is a moving point on segment $$B E$$, and let $$P^{'}$$ be the reflection of point $$P$$ across line $$D E$$. Point $$Q$$ is a moving point on $$A C$$, and connect $$P^{'} Q$$ and $$D Q$$. If $$A E = 9$$ and $$C E = 7$$, then the maximum value of $$D Q - P^{'} Q$$ is      .

Answer: $$\frac{8 \sqrt{2}}{3}$$/$$\frac{8}{3} \sqrt{2}$$


As shown in the figure, the side length of square $$ABCD$$ is 4, and points E and F lie on sides $$AD$$ and $$BC$$, respectively. Folding quadrilateral $$ABFE$$ along $$EF$$ yields quadrilateral $$EFNM$$, and the image of point A, denoted M, lies exactly on line $$CD$$. If $$DM = 1$$, then the length of segment $$BF$$ is    .

Answer: $$\frac{9}{8}$$ or $$\frac{25}{8}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = \sqrt{2}, A D = 2$$, $$E$$ is the midpoint of side $$A D$$, point $$F$$ lies on side $$C D$$, connect $$E F$$, fold $$\triangle D E F$$ along $$E F$$, and let the image of point $$D$$ be $$D^{'}$$, connect $$B D^{'}$$. If $$B D^{'} = 2$$, then $$D F =$$      .

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


As shown in the figure, the three vertices $P$, $Q$, and $R$ of the equilateral triangle $PQR$ lie on three sides $AD$, $AB$, and $DC$ of the square $ABCD$, respectively. Given that the side length of square $ABCD$ is $4\sqrt{6}$, the sum of the lengths of $AQ$ and $DR$ is             .

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


As shown in the figure, in square $$A B C D$$, a semicircle $$O$$ is drawn with $$B C$$ as its diameter, and an arc $$\overset{⌢}{A C}$$ is drawn with center $$D$$ and radius $$D A$$, intersecting the semicircle $$O$$ at point $$P$$. We call point $$P$$ a "wonderful point" of square $$A B C D$$. Many line segments passing through the wonderful point have beautiful properties, both in terms of position and quantity, worth exploring. Connect $$P A$$, $$P B$$, $$P C$$, $$P D$$, and extend $$P D$$ to intersect $$A B$$ at point $$F$$. Among the following conclusions: ① $$F D = F B + B C$$; ② $$\angle A P C = 135 \circ$$; ③ $$S_{\triangle P B C} = \frac{1}{2} A P^{2}$$; ④ $$\text{tan} \angle B A P = \frac{1}{3}$$; the correct conclusion(s) is/are      .

Answer: ①②③④


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 2$$, point $$P$$ is a point on side $$A B$$, and point $$A$$ is symmetric to point $$A^{'}$$ with respect to line $$P D$$. The following conclusions: ① When line $$A^{'} D$$ divides the area of the rectangle in the ratio $$1 : 3$$, $$A P = 2 \sqrt{2} - 2$$; ② When points $$P$$, $$A^{'}$$, and $$C$$ are collinear, $$A P = 4 - 2 \sqrt{3}$$; ③ Let the length of segment $$A^{'} B$$ be $$d$$, then $$2 \sqrt{5} - 2 \leq d \leq 2 \sqrt{5} + 2$$; ④ Let the area of $$\triangle A^{'} A B$$ be $$S$$, then $$0 < S \leq \frac{32}{5}$$. Among these, the correct ones are      . (Only fill in the serial numbers)

Answer: ②④


In quadrilateral $$A B C D$$, $$\angle A = \angle D = 90 \circ$$, $$C D = B C$$, the following four judgments:

① If $$\angle C = 120 \circ$$, then $$A D = \frac{1}{2} B C$$;
② Connect $$A C$$, $$D B$$, if $$A C$$ bisects $$D B$$ perpendicularly, then $$A D = B C$$;
③ Connect $$A C$$, construct $$\angle D A C = \angle A C B$$, then quadrilateral $$A B C D$$ is a square;
④ The reflection of point $$A$$ over line $$B D$$ must lie on line $$B C$$.
Among these, the correct serial numbers are        . (Write all correct serial numbers)

Answer: ②③④


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, construct squares $$A B E F$$, $$B C G H$$, and $$A C M N$$ outwardly on the three sides of the triangle. Draw $$B I \bot E H$$ with foot at point I, and extend $$I B$$ to intersect $$A C$$ at point J. The following conclusions are given: ① $$A B = M G$$.② $$S_{\triangle B E H} = S_{\triangle A F N}$$.③ Draw $$B I \bot E H$$ with foot at point I, and extend $$I B$$ to intersect $$A C$$ at point J, then $$A J = C J$$.④ If J is the midpoint of $$A C$$, then $$2 B J = E H$$.Which of the above conclusions are correct? (Only fill in the serial numbers)

Answer: ①②③④


As shown in the figure, the side length of square $$A B C D$$ is $$2 \sqrt{5}$$, point $$E$$ is the midpoint of $$C D$$, $$B E$$ intersects $$A C$$ at point $$M$$, $$F$$ is a point on $$A D$$, connect $$B F$$ which intersects $$A C$$ and $$A E$$ at points $$G$$ and $$H$$ respectively, and $$B F \bot A E$$, connect $$M H$$, then $$M H =$$        .

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


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, $$B C = 9$$, point D is a point on side $$B C$$ such that $$B D = 6$$, draw $$D E \bot A B$$ with foot at point E, connect $$A D$$, then $$A D =$$    ; point F is the midpoint of $$A D$$, connect $$C F$$, draw $$F G \bot C F$$ intersecting $$D E$$ at point G, then $$F G =$$     .

Answer: $$3 \sqrt{7}$$     $$\frac{\sqrt{39}}{2}$$


As shown in the figure, in the square $$ABCD$$ with side length 4, point $$E$$ is the midpoint of $$BC$$, and point $$F$$ is a moving point on $$AB$$. Fold $$\triangle BEF$$ along $$EF$$ to obtain $$\triangle GEF$$. Connect $$GC$$, and construct $$\triangle GHC$$ as the reflection of $$\triangle GEC$$ across $$GC$$. Connect $$DG$$ and $$DH$$. When $$\triangle DGH$$ is an isosceles triangle, the length of $$BF$$ is        .

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


As shown in the figure, in rectangle $$A B C D$$, $$A D = 3$$, $$A C = 6$$, point $$E$$ is the midpoint of $$A B$$, point $$F$$ is a point on diagonal $$A C$$, $$\triangle G E F$$ is symmetric to $$\triangle A E F$$ with respect to line $$E F$$, $$E G$$ intersects $$A C$$ at point $$H$$, when one of the interior angles of $$\triangle C G H$$ is $$90 \circ$$, then the length of $$C G$$ is      .

Answer: $$\frac{3 \sqrt{3}}{2}$$ or $$\frac{3 \sqrt{7}}{2}$$


As shown in the figure, in square $$ABCD$$, $$AB = 4$$, point E is a moving point on $$AB$$, point F is on $$CD$$, and $$AE = CF$$. From point B, draw $$BG \bot EF$$ intersecting $$AD$$ at point G, with foot at point M:

(1) When point G is the midpoint of $$AD$$, the length of $$AE$$ is            ; (2) Connect $$AM$$, then the minimum value of $$AM$$ is           ;

Answer: 1     $$\sqrt{10} - \sqrt{2}$$/$$- \sqrt{2} + \sqrt{10}$$


As shown in the figure, in square $$ABCD$$, point $$E$$ is a point on side $$BC$$, connect $$DE$$, point $$F$$ is the midpoint of $$DE$$, draw a perpendicular to $$DE$$ through point $$F$$, intersecting $$AB$$ and $$CD$$ at points $$M$$ and $$N$$ respectively, connect $$AC$$ intersecting $$MN$$ at point $$G$$. If $$\angle DNG = 60^\circ$$ and $$AB = 3$$, then the length of $$FG$$ is $$      $$

Answer: $$\sqrt{3}$$


As shown in the figure, in rectangle $$A B C D$$, $$B C = 6$$, points $$E$$ and $$H$$ are the midpoints of sides $$A D$$ and $$B C$$ respectively. Connect $$B E$$, and let point $$F$$ be a moving point on $$B E$$. Connect $$H F$$ and $$D F$$; extend $$D F$$ to intersect $$A B$$ at point $$P$$. If $$P B = P F$$, then the length of $$H F$$ is        .

Answer: 3


As shown in the figure, O is the intersection point of the diagonals of rectangle $$A B C D$$, point E lies on side $$A D$$, connect $$O E$$, rotate segment $$O E$$ counterclockwise around point O by $$90 \circ$$ to obtain segment $$O F$$ (point F lies inside rectangle $$A B C D$$), connect $$A F$$ and $$E F$$. If $$A B = 2$$, $$A D = 4$$, then the maximum area of $$\triangle A E F$$ is       .

Answer: $$\frac{9}{8}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$D$$ is the midpoint of $$A B$$, and connect $$C D$$. Fold $$\triangle B C D$$ along $$C D$$ to obtain $$\triangle E C D$$, and connect $$A E$$. If $$A C \bot D E$$ at point $$F$$, and $$B C = 4$$, then the length of $$A F$$ is        .

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


As shown in the figure, the square $$ABCD$$ has side length 4, and points $$E$$ and $$F$$ lie on sides $$BC$$ and $$CD$$ respectively. $$AE$$ bisects $$\angle BAC$$. Connect $$BF$$, intersecting $$AE$$ and $$AC$$ at points $$G$$ and $$H$$ respectively, and $$AE = BF$$. The following four conclusions are given: ① $$AE$$ bisects $$BH$$ perpendicularly; ② If point $$P$$ is a moving point on side $$AB$$, then the minimum value of $$PH + PC$$ is $$4\sqrt{3}$$; ③ $$GH^{2} = AG \cdot EG$$; ④ $$S_{\triangle ABH} = 6\sqrt{2}$$. Among these, the correct ones are             .

Answer: ①②③


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, $$A B = 4$$, point D moves along side $$A C$$ from C to A, point E moves along side $$B C$$ from B to C, and $$C D = B E$$. Connect $$B D$$ and $$A E$$, intersecting at point P. Rotate side $$A C$$ clockwise 90° about point C to obtain $$C M$$, and on ray $$C M$$, take segment $$C F$$ such that $$C F = \sqrt{3} A C$$. During the motion of D and E, find the minimum value of $$\frac{1}{2} P C + P F$$      .

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


As shown in the figure, in $$R t \triangle A B C$$, $$\angle C = 90 \circ$$, $$B C = 4$$, $$A B = 8$$, point D is a moving point on side $$A C$$, and a square $$B D E F$$ is constructed with $$B D$$ as a side above $$B D$$. Then the minimum value of $$A E$$ is                , and at this time the length of $$C D$$ is                   .

Answer: $$6 - 2 \sqrt{3}$$/$$- 2 \sqrt{3} + 6$$     $$2 \sqrt{3} - 2$$/$$- 2 + 2 \sqrt{3}$$


As shown in the figure, in parallelogram $$A B C D$$, $$\angle A B C = 120 \circ$$, $$A B = 10$$. Connect $$B D$$, and $$B D \bot C D$$. Let $$C E$$ bisect $$\angle D C B$$ and intersect $$A D$$ at point $$E$$. Point $$N$$ lies on side $$B C$$, and $$B C = 4 C N$$. If segment $$P Q$$ (with point $$P$$ to the left of point $$Q$$) moves along segment $$C E$$, with $$P Q = 5 \sqrt{3}$$, connect $$B P$$ and $$N Q$$. Find the minimum value of $$B P + P Q + Q N$$.

Answer: $$5 \sqrt{7} + 5 \sqrt{3}$$


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, $$D$$ is a point on side $$B C$$, and connect $$A D$$. Construct $$\text{Rt} \triangle A E C$$ with hypotenuse $$A C$$ to the right of side $$A C$$, and connect $$B E$$. Let $$F$$ be a point on $$B E$$ such that $$B F = \frac{2}{3} B E$$, and connect $$D F$$. If $$A B = 6$$ and $$C D = 1$$, when $$D F$$ takes its minimum value, the area of $$\triangle B D F$$ is           .

Answer: $$\frac{5 \sqrt{3}}{2} - \frac{5 \sqrt{21}}{7}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 60 \circ$$, point $$D$$ lies on $$A B$$, $$C D = 14$$, $$\angle B D C = 60 \circ$$, extend $$C B$$ to point $$E$$ such that $$C E = A C$$, draw $$E F \bot C D$$ through point $$E$$, intersecting $$C D$$ at point $$F$$ and intersecting $$A B$$ at point $$G$$, if $$2 D G = A D$$, then $$D F =$$        .

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


As shown in the figure, $$\triangle A B C \sim \triangle B D C$$, $$\angle A = \angle C B D = 90 \circ$$, $$A B = 4$$, $$B E$$ bisects $$\angle C B D$$ and intersects $$C D$$ at point E, and intersects the extension of $$A C$$ at point F. Connect $$D F$$. If the triangle with vertices B, C, E is similar to $$\triangle D E F$$, then $$A C =$$      .

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


As shown in the figure, two congruent right triangles $$BCE$$ and $$DAF$$ are constructed outwardly with $$BC$$ and $$AD$$ as the hypotenuses of square $$ABCD$$. From point C, draw $$CG \bot AF$$ intersecting at point G and intersecting $$AD$$ at point H. From point B, draw $$BI \bot CG$$ intersecting at point I. From point D, draw $$DK \bot BE$$ intersecting the extension of $$EB$$ at point K and intersecting $$CG$$ at point L. If $$S_{\text{quadrilateral } ABI G} = 2 S_{\triangle BCE}$$ and $$GH = 1$$, then the length of $$DK$$ is    .

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


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle B = \angle D = 90 \circ$$, $$A B = 2$$, $$A D = 3$$, points $$M$$, $$N$$ lie on sides $$B C$$, $$C D$$ respectively. When $$\angle A M N + \angle A N M = 120 \circ$$, the perimeter of $$\triangle A M N$$ is minimized. Find its minimum perimeter.

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


As shown in the figure, it is given that quadrilateral $$ABCD$$ is a rectangle, $$AB = 8$$, $$AD = 6$$, $$F$$ is a moving point on side $$BC$$, $$O$$ is the midpoint of $$AC$$, $$OE \bot OF$$ intersects $$AB$$ at $$E$$, and connect $$EF$$, $$OB$$. If $$OB$$ divides the area of $$\triangle OEF$$ into two parts in the ratio $$1:2$$, then the length of $$BF$$ is          .

Answer: $$\frac{75}{41}$$ or $$\frac{75}{17}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$\angle A B C = 90 \circ$$, connect $$B D$$, $$A B = A D = 2$$, $$B D = 2 \sqrt{2}$$, points $$E$$, $$F$$ lie on sides $$B C$$, $$C D$$ respectively, and $$D F = C E$$, connect $$B F$$, $$D E$$, if $$C D = 3$$, then the minimum value of $$B F + D E$$ is         .

Answer: $$\sqrt{29}$$


As shown in the figure, in rectangle $$A B C D$$, $$A D = 17$$, $$A B = 16$$, point E is a moving point on the sides of the rectangle. Connect $$D E$$, fold $$\triangle A D E$$ along the line $$D E$$, and point A lands at point F (point F is below the line $$A D$$). Connect $$C F$$. When $$\triangle C D F$$ is an isosceles triangle with $$D F$$ as the leg, the value of $$\tan \angle A D E$$ is      .

Answer: 4 or $$\frac{1}{4}$$


As shown in the figure, in rectangle $$A B C D$$, $$A D = 2$$, $$A B = 4$$, $$E$$ and $$F$$ are moving points on sides $$A B$$ and $$C D$$ respectively, and $$E F \bot A C$$. Then the minimum value of $$A F + C E$$ is          .

Answer: 5


As shown in the figure, the radius of ⊙O is 1, point B is the midpoint of radius OA, and point C is a moving point on ⊙O. Rotate CB clockwise by 90° around point B to obtain BD. Let M be the midpoint of AD. Then the maximum value of OM is ______.

Answer: $$\frac{1}{2} + \frac{\sqrt{10}}{4}$$


As shown in the figure, $$\triangle A B C$$ and $$\triangle A D E$$ are both equilateral triangles. Connect $$C E$$ and $$B E$$.

(1) If $$A E = 4$$, then the area of $$\triangle A D E$$ is      ; (2) If $$B C \bot C E$$ and $$B E = 4$$, then the maximum area of $$\triangle A D E$$ is      .

Answer: $$4 \sqrt{3}$$;     $$6 + 4 \sqrt{3}$$/$$4 \sqrt{3} + 6$$.


As shown in the figure, given $$\angle M C N = 30 \circ$$, if points $$E$$ and $$A$$ lie on ray $$C M$$, and satisfy $$A C = 6 \sqrt{3}$$, $$E C : A E = 2 : 1$$, and $$G$$ is a moving point on ray $$C N$$, simultaneously construct $$\angle G E H = 30^\circ$$ on the right side of $$E G$$, and satisfy $$E G = \sqrt{3} E H$$, then the area of $$\triangle C E H$$ is     . If the trajectory of point $$H$$ intersects ray $$C N$$ at point $$Q$$, when $$2 A H + H Q$$ attains its minimum value, the value of $$A H$$ is     .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$\angle A B C = 30 \circ$$, an equilateral triangle $$\triangle D C B$$ is constructed on side $$B C$$ above it. Points $$E$$ and $$F$$ are two moving points on sides $$A C$$ and $$A B$$ respectively, such that $$A E = B F$$. If $$B C = 6$$, then the minimum value of $$D E + D F$$ is           .

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


As shown in the figure, the diameter $$AB$$ of $$\bigodot O$$ intersects the chord $$CD$$ at point E, and point P is a point on $$CD$$ such that $$\angle APB = 120^\circ$$. If $$AB = 6$$ and $$CD = 4$$, then the maximum value of $$PA \cdot PB$$ is     , and the maximum value of $$PA + PB$$ is        .

Answer: 8     $$2 \sqrt{11}$$


As shown in the figure, $$BC$$ is the diameter of $$\bigodot O$$, point A lies on $$\bigodot O$$, $$AD \bot BC$$, with foot at D, $$\overset{⌢}{AE} = \overset{⌢}{AB}$$, the extensions of $$BE$$ and $$AC$$ intersect at point G, the extension of $$AD$$ intersects $$BE$$ at point F. If $$BG = 10$$ and $$BD - DF = 1$$, then $$AB =$$        .

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


As shown in the figure, in the rhombus $$A B C D$$, $$\angle B = 120 \circ$$, point E is a point on $$B C$$, connect $$A E$$, rotate $$A E$$ clockwise around point E by $$120 \circ$$ to $$F E$$, connect $$A F$$ intersecting $$C D$$ at point G, if $$\frac{D G}{C G} = \frac{1}{4}$$, then $$\frac{B E}{C E} =$$      .

Answer: $$\frac{4}{3}$$/$$1 \frac{1}{3}$$


Given rectangle $$A B C D$$, connect diagonal $$A C$$, point $$F$$ lies on side $$C D$$, connect $$A F$$, point $$E$$ is the midpoint of $$A F$$, connect $$D E$$, if $$\angle D E F = \angle B A C, D E = \sqrt{5}, C F = 1$$, then the length of $$A B$$ is      .

Answer: 3


As shown in the figure, the side length of square $$A B C D$$ is 4, and E is a point on side $$C D$$. Connect $$A E$$, and from point B, draw $$B F \bot A E$$ intersecting at point F. Point G is the reflection of point F across line $$C D$$, and H is the midpoint of $$C G$$. Then the minimum value of $$A H$$ is            .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = \sqrt{5}$$, $$B C = 2$$, point $$D$$ is any moving point on $$B C$$ (not coinciding with $$B$$ or $$C$$). From point $$B$$, draw $$B H \bot A D$$, with foot at point $$H$$, and connect $$C H$$. Then the minimum value of $$C H$$ is        .

Answer: $$\frac{\sqrt{13} - \sqrt{5}}{2}$$


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle B A C = 120 \circ$$, point $$D$$ is a point on side $$A B$$ (not coinciding with point $$B$$), connect $$C D$$, rotate segment $$C D$$ counterclockwise by $$90 \circ$$ about point $$D$$, and let the image of point $$C$$ be point $$E$$, connect $$B E$$. If $$A B = 10$$, then the maximum area of $$\triangle B D E$$ is       .

Answer: $$\frac{225}{8}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$B C = 6$$, point E is the midpoint of side $$B C$$, connect $$A E$$, points F and G lie on $$A E$$ and $$D E$$ respectively, and $$F G \parallel A D$$, the reflection of point E over $$F G$$ is $$E^{'}$$, $$A D$$ intersects $$E^{'} F$$ and $$E^{'} G$$ at H and I respectively. If $$A H + D I = H I$$, then the area of quadrilateral $$F E G E^{'}$$ is                    .

Answer: $$\frac{27}{2}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$A D = 10$$. A moving point $$E$$ lies on side $$A B$$, and an arc is drawn with center at point $$E$$ and radius $$B E$$. Point $$G$$ is a moving point on this arc. If $$A E = 1$$, connect $$C G$$ and $$D G$$, and let $$P$$ and $$Q$$ be the midpoints of $$D G$$ and $$C G$$, respectively. Connect $$P Q$$, and let $$M$$ be the midpoint of $$P Q$$. Then the minimum value of $$C M$$ is      .

Answer: $$\frac{2 \sqrt{41} - 5}{2}$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle C = 90 \circ$$, $$A C = 6$$, $$B C = 8$$. Point $$D$$ is the midpoint of the hypotenuse $$A B$$, and point $$P$$ is a moving point on side $$A C$$. Connect $$P D$$, and rotate segment $$P D$$ clockwise by $$90 \circ$$ about point $$D$$ to obtain segment $$P^{'} D$$. Segment $$P^{'} D$$ intersects side $$B C$$ at point $$Q$$. When $$\triangle B Q P^{'}$$ is a right triangle, the length of $$A P$$ is      .

Answer: 3 or 5/5 or 3


As shown in the figure, there is a moving point E on side BC of rectangle ABCD. Connect AE, and construct rectangle AEGF with AE as one side, such that side FG passes through point D. If AB = √3 and BC = 4, when triangle AED is an isosceles triangle, the length of BE is        .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle C A B = 90 \circ$$. Rotate side $$C A$$ counterclockwise about point $$C$$ by $$90 \circ$$ to obtain segment $$C E$$, and rotate side $$C B$$ clockwise about point $$B$$ by $$90 \circ$$ to obtain segment $$B D$$. Let $$C D$$ intersect $$A E$$ at point $$F$$. If $$E F = \sqrt{2}$$ and $$D F = \sqrt{10}$$, then the length of side $$A B$$ is          .

Answer: 2


As shown in the figure, there is a rectangular paper $$A B C D$$, with $$A B = 7$$, $$B C = 3$$. Points $$M$$ and $$N$$ lie on sides $$A B$$ and $$C D$$ respectively, and $$C N = 1$$. The quadrilateral $$B C M N$$ is folded along $$M N$$, causing points $$B$$ and $$C$$ to land on points $$B^{'}$$ and $$C^{'}$$ respectively. As point $$M$$ moves from point $$A$$ to point $$B$$, if side $$M B^{'}$$ intersects side $$C D$$ at point $$E$$, then the length of the path traced by point $$E$$ is      .

Answer: $$\sqrt{10} - \frac{9}{4}$$


As shown in the figure, $$\triangle A B C$$ is an equilateral triangle, $$D$$ is the midpoint of side $$B C$$, point $$E$$ lies on side $$A C$$, connect $$D E$$, fold $$\triangle C D E$$ along $$D E$$ by $$180^\circ$$ to obtain $$\triangle F D E$$, connect $$B E$$ and $$B F$$, given that $$A C = \sqrt{10} B F$$, $$S_{\triangle B E F} = \frac{3}{2}$$, then the area of $$\triangle A B C$$ is      .

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


As shown in the figure, in rectangle ABCD, $$AB = 4$$, $$BC = 3$$, translate $$\triangle BCD$$ along ray BD by length $$a \left(\right. a > 0 \left.\right)$$ to obtain $$\triangle B^{'} C^{'} D^{'}$$, connect $$AB^{'}$$, $$AD^{'}$$, then when $$\triangle AB^{'} D^{'}$$ is a right triangle, the length of $$a$$ is           .

Answer: $$\frac{7}{5}$$ or $$\frac{16}{5}$$


In $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C$$, point $$D$$ is a point on side $$A B$$, through point $$D$$ draw $$D E \bot A B$$, intersecting $$B C$$ at point $$E$$, connect $$A E$$, take the midpoint $$P$$ of $$A E$$, connect $$D P$$, $$C P$$. If $$B C = 3 B D = 6 \sqrt{2}$$, rotate $$\triangle B D E$$ around point $$B$$ freely in the plane, when $$B E \bot A B$$, then the length of segment $$C P =$$        .

Answer: $$8$$ or $$4$$


As shown in the figure, in square $$ABCD$$, $$E$$ is the midpoint of $$AB$$. Rotate $$DE$$ clockwise around point $$E$$ by $$90^\circ$$ to obtain $$EF$$. Connect $$CF$$ and $$AF$$, and let $$AF$$ intersect $$DE$$ at point $$M$$. If $$CF = 6$$, then the length of $$DM$$ is        .

Answer: $$\frac{18 \sqrt{10}}{7}$$/$$\frac{18}{7} \sqrt{10}$$


As shown in the figure, in square $$ABCD$$, point M is a point on side $$CD$$. Connect $$AM$$. Rotate $$\triangle ADM$$ clockwise about point $$A$$ by $$90^\circ$$ to obtain $$\triangle ABN$$. On $$AM$$ and $$AN$$, respectively, take points $$E$$ and $$F$$ such that $$AE = AF = BC$$. Connect $$EF$$, intersecting diagonal $$BD$$ at point $$G$$. Connect $$AG$$ and extend it to intersect $$BC$$ at point $$H$$. If $$AM = \frac{25}{3}$$ and $$CH = 2$$, then the length of $$AG$$ is ________.

Answer: $$\frac{40}{7}$$/$$5 \frac{5}{7}$$


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4$$, $$A D = 6$$, points P, M, N move along sides $$A B$$, $$A D$$, $$B C$$ respectively, and segment $$M N$$ always passes through the center of symmetry of the rectangle. The minimum perimeter of $$\triangle P M N$$ is            .

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


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$A B = A C = 8$$, $$B O = \frac{1}{4} A B$$, point $$M$$ is a moving point on side $$B C$$, rotate segment $$O M$$ counterclockwise around point $$O$$ by $$90^\circ$$ to obtain $$O N$$, and connect $$A N$$ and $$C N$$. (1) When point $$N$$ lies on $$A B$$, $$A N =$$     ; (2) The minimum perimeter of $$\triangle C A N$$ is                    .

Answer: 4     $$8 + 4 \sqrt{10}$$


As shown in the figure, it is given that quadrilateral $$A B C D$$ is a rhombus, $$A B = 4$$, $$\angle C = 60 \circ$$, $$B D$$ is a diagonal, and E is a moving point on side $$C D$$, with $$E F \parallel B D$$ intersecting $$B C$$ at point F. Connect $$A E$$ and $$A F$$, and let G be the midpoint of $$A E$$. Connect $$F G$$.
① If E is the midpoint of $$D C$$, then the length of $$C F$$ is     ;
② As point E moves, the minimum value of $$G F$$ is       .

Answer: $$2$$     $$3$$


As shown in the figure, in the rhombus $$A B C D$$, $$A B = 6$$, $$E$$ is a trisection point of side $$C D$$, $$\angle D = 60^\circ$$, fold $$\triangle A D E$$ along $$A E$$ to obtain $$\triangle A F E$$, and let line $$E F$$ intersect $$B C$$ at point $$P$$. Then $$P C =$$       .

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


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 30 \circ$$, $$A B = A C = 2$$, point E is a moving point on ray $$A C$$, $$D E \parallel A B$$, and $$D E = 2$$. When the value of $$A D + B D$$ is minimized, the measure of $$\angle D B C$$ is     .

Answer: $$45 \circ$$


As shown in the figure, in $$\triangle A B C$$, $$\angle B A C = 45 \circ$$, $$C D \bot A B$$ at point $$D$$, $$A E \bot B C$$ at point $$E$$, $$A E$$ intersects $$C D$$ at point $$F$$, connect $$B F$$ and $$D E$$, among the following conclusions: ① $$A F = B C$$; ② $$2 \cos \angle D E B = \frac{1}{2}$$, ③ $$A E - C E = \sqrt{2} E D$$, ④ if $$\angle C A E = 30 \circ$$, then $$\frac{A F + B F}{A C} = 1$$, the correct ones are      . (fill in the serial numbers).

Answer: ①③④


As shown in the figure, in square $$ABCD$$, $$AB = 4\sqrt{2}$$, diagonals $$AC$$ and $$BD$$ intersect at point $$O$$. Point $$E$$ is a point on diagonal $$AC$$, connect $$BE$$, draw $$EF \bot BE$$, intersecting $$BD$$ and $$CD$$ at points $$F$$ and $$G$$ respectively, connect $$BG$$ intersecting $$AC$$ at point $$H$$. Fold $$\triangle EGH$$ along $$EG$$, and the image point $$P$$ of point $$H$$ falls exactly on $$BD$$, forming $$\triangle EPG$$. If point $$G$$ is the midpoint of $$CD$$, then the area of $$\triangle PFG$$ is        .

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


As shown in the figure, in quadrilateral $$A B C D$$, $$A D \parallel B C$$, $$A C$$ and $$B D$$ are diagonals, $$A C \bot C D$$, $$A B = A C$$. If $$\angle A B D = 2 \angle A D C$$ and $$C D = 2 \sqrt{5}$$, then the length of $$A D$$ is            .

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


As shown in the figure, in quadrilateral ABCD, diagonals AB and CD intersect at point E, $$CA = CB = CD$$, $$\angle ACB = 90^\circ$$, if $$S_{\triangle ACE} - S_{\triangle BDE} = 8$$, then the length of segment AD is         .

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


Given in rectangle $$A B C D$$, $$A D = 9$$, $$A B = 12$$, O is the center of the rectangle; in $$\text{Rt} \triangle A E F$$, $$\angle E A F = 90 \circ$$, $$A E = 6$$, $$A F = 8$$. Rotate $$\triangle A E F$$ around point A in a clockwise direction for one full turn, then the height from side $$E F$$ is      . Connect $$C E$$, take the midpoint M of $$C E$$, connect $$F M$$, write the range of values for $$F M$$      .

Answer: $$4.8$$     $$\frac{2 \sqrt{73} - 15}{2} \leq F M \leq \frac{15 + 2 \sqrt{73}}{2}$$


As shown in the figure, in square $$ABCD$$ with side length 4, point E is a moving point on side $$BC$$ (point E does not coincide with B or C). Connect $$AE$$. From point B, draw $$BF \bot AE$$ intersecting at point F. Point G is the reflection of point C across line $$BF$$. Connect $$AG$$, $$DG$$, and $$GF$$. When $$GF$$ attains its minimum value, the area of $$\triangle AGD$$ is            .

Answer: $$8 - \frac{16 \sqrt{5}}{5}$$


As shown in the figure, in the equilateral triangle $$A B C$$, $$A D = B E = C F$$, $$A N = C M = B P$$, $$D E = 7$$, $$M N = 5$$, and $$M N \parallel D E$$, then the length of $$A B$$ is          .

Answer: $$\frac{35 \sqrt{13}}{13}$$


As shown in the figure, in $$\triangle A B C$$, $$A C = B C$$, $$\angle A C B = 90 \circ$$, point D lies on side $$A C$$, $$A D = 2 C D$$, an equilateral triangle $$\triangle B D E$$ is constructed with $$B D$$ as a side, and $$D E$$ intersects $$A B$$ at point F. Then $$\frac{E F}{F D} =$$      .

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


As shown in the figure, quadrilateral $$A B C D$$ is a square with side length $$1 \text{cm}$$, point F is a moving point on the extension of side $$B C$$, point E is above line $$B C$$, and $$E F = B F$$, $$E F \bot B F$$. If $$\triangle D E F$$ is exactly an isosceles triangle, then the length of $$C F$$ is      $$\text{cm}$$.

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


As shown in the figure, point D is a moving point on the side BC of the equilateral triangle ABC, with AB = 2. Construct an equilateral triangle ADE on the right side of AD. AC intersects DE at point F. Which of the following conclusions are correct? (Fill in the serial numbers) ① ∠ACD = ∠ACE; ② AF · CF = DF · EF; ③ The maximum area of triangle CDE is √3/2; ④ The minimum length of segment DE is √3.

Answer: ①②④


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = \sqrt{7}$$, $$B C = 2 \sqrt{3}$$, $$P$$ is a point inside $$\triangle A B C$$, $$\angle A P B = \angle A P C = 120^\circ$$, then the value of $$P A + P B + P C$$ is      .

Answer: 5


As shown in the figure, in square $$ABCD$$, $$AB = 8$$, point $$P$$ is any point on ray $$BC$$ (not coinciding with points $$B$$ and $$C$$). Connect $$AP$$, and construct square $$APGH$$ on the right side of $$AP$$. Connect $$AG$$, intersecting ray $$CD$$ at point $$E$$. When the length of $$ED$$ is 2, the length of $$BP$$ is        .

Answer: $$\frac{24}{5}  or  \frac{40}{3}$$


As shown in the figure, in quadrilateral $$A B C D$$, $$B D$$ is a diagonal, and $$E$$ is a point on side $$A D$$. Connect $$C E$$ intersecting $$B D$$ at point $$O$$. If $$\angle A = \angle B C D = \angle B O C = 120 \circ$$, $$A D = \frac{69}{2}$$, $$A B = 12$$, $$\frac{B C}{C D} = \frac{4}{3}$$, then the value of $$\frac{B D}{C E}$$ is      .

Answer: $$\frac{23}{12}$$/$$1 \frac{11}{12}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A B C = 90 \circ$$, $$A B = 8$$, point P is a moving point on side $$A B$$, and a circle with diameter $$B P$$ intersects $$C P$$ at point Q. If the minimum length of segment $$A Q$$ is 4, then the area of $$\triangle A B C$$ is     .

Answer: 48


As shown in the figure, $$CD$$ is the median on the hypotenuse $$AB$$ of $$\text{Rt} \triangle ABC$$, point $$E$$ is the midpoint of $$CD$$, connect $$AE$$, point $$F$$ is the midpoint of $$AE$$, connect $$DF$$, if $$DE = DF = 2$$, then the length of segment $$AC$$ is           .

Answer: $$2 \sqrt{10}$$ or $$2 \sqrt{6}$$/$$2 \sqrt{6}$$ or $$2 \sqrt{10}$$


As shown in the figure, in $$\triangle ABC$$, $$AC = AB$$, $$\angle BAC = 90^\circ$$, $$D$$ is a point on side $$AC$$, connect $$BD$$, $$AF \perp BD$$ at point $$F$$, point $$E$$ lies on $$BF$$, connect $$AE$$ and $$CE$$, $$\angle EAF = 45^\circ$$, if $$\tan \angle ECD = \frac{3}{4}$$, $$BC = 6$$, then the length of $$BE$$ is  .

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


As shown in the figure, the diagonals of rhombus $$A B C D$$ intersect at point O. Points M and N are moving points on segments $$A B$$ and $$A C$$, respectively, such that $$A M = C N$$. If $$A B = 5$$ and $$B D = 6$$, then the minimum value of $$D M + D N$$ is      .

Answer: $$\frac{13 \sqrt{10}}{5}$$


As shown in the figure, $$\triangle A B C$$ is an isosceles right triangle, D is the midpoint of $$A B$$, E and F are points on $$A C$$ and $$B C$$ respectively such that $$D F \bot D E$$, given $$A E = 2$$, $$C E = 5$$, M is a point on $$B C$$, connect $$M E$$, and satisfies $$\angle C M E = 2 \angle A D E$$, then $$E M = \\$$           .

Answer: $$\frac{29}{4}$$


As shown in the figure, in right triangle $$A B C$$, $$\angle A B C = 90 \circ, A B = 8, B C = 6$$, the angle bisectors of $$\angle C A B$$ and $$\angle A C B$$, namely $$A E, C D$$, intersect at point P, the altitude from $$B$$ to side $$A C$$, denoted as $$B F$$, intersects $$A E$$ and $$C D$$ at points G and H respectively, M and N are the midpoints of $$D H$$ and $$E G$$ respectively, connect $$M N$$, $$B M$$, $$B N$$, which of the following statements is correct                    .
①$$B F = 4.8$$, ② the ratio of the areas of $$\triangle A B P$$ and $$\triangle C B P$$ is $$3 : 4$$, ③ $$\triangle B D H$$ is an isosceles triangle, ④ $$B N \bot A E$$, ⑤ $$\angle M N P = \angle E A B$$ (please fill in the corresponding serial numbers)

Answer: ①③④⑤


Given that the vertices M, N, P, Q of rectangle $$M N P Q$$ lie respectively on the sides $$D E$$, $$F A$$, $$A B$$, $$C D$$ of regular hexagon $$A B C D E F$$, and as point M moves from E to D, the following judgments about rectangle $$M N P Q$$:
① The area and perimeter of rectangle $$M N P Q$$ remain constant;
② The area of rectangle $$M N P Q$$ gradually decreases;
③ The perimeter of rectangle $$M N P Q$$ gradually increases;
④ The length of the diagonal of rectangle $$M N P Q$$ has a minimum value.
The correct one(s) is/are    . (Fill in the serial number)

Answer: ②④/④②


As shown in the figure, in $$\triangle A B C$$, point D is a point on $$B C$$, $$\angle A D C = 60 \circ$$, point E lies on segment $$A D$$, $$\angle B E C = 120 \circ$$, if $$B C = 3 \sqrt{3}$$, $$A E = 2 \sqrt{3}$$, then the maximum value of $$A C$$ is        .

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


As shown in the figure, the square $$A B C D$$ has a side length of 6, point E is the midpoint of $$B C$$, connecting $$A E$$ intersects the diagonal $$B D$$ at point G, connecting $$C G$$ and extending it intersects $$A B$$ at point F, connecting $$D E$$ intersects $$C F$$ at point H, and connecting $$A H$$. The following conclusions: ① $$C F \bot D E$$; ② $$\frac{C H}{H F} = \frac{2}{3}$$; ③ $$A D = A H$$; ④ $$G H = \frac{3}{5} \sqrt{5}$$, among which the correct conclusions are      .

Answer: ①②③


As shown in the figure, in the isosceles right triangle $$\text{Rt} \triangle A B C$$, $$\angle B A C = 90 \circ$$, points D and E lie on segments $$A B$$ and $$A C$$ respectively, with $$A D = A E$$; BE and CD intersect at point N; AF ⊥ BE intersects BC at point F; FG ⊥ CD intersects AC at point M and intersects the extension of BE at point G. The following statements: ① $$\angle A B E = \angle F A C$$; ② $$G E = M E$$; ③ $$B G = A F + F G$$; ④ $$C_{\triangle A F M} = B E + C M$$; ⑤ $$S_{\triangle B D N} : S_{\triangle A F C} = C E : A C$$. Among these, the correct ones are      .

Answer: ①③⑤


As shown in the figure, in rectangle $$A B C D$$, $$A B = 2$$, point $$E$$ is a moving point on $$A C$$, connect $$D E$$, draw line $$l$$ perpendicular to $$A C$$ passing through point $$C$$, draw $$D F \bot D E$$ intersecting $$l$$ at point $$F$$, draw $$D G \bot E F$$ at point $$G$$, $$t a n \angle E D G = \sqrt{2}$$, point $$H$$ is the midpoint of $$A D$$, connect $$H E$$, then the minimum value of $$H E + \frac{\sqrt{6}}{3} E C$$ is   .

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


As shown in the figure, in square ABCD, point E is connected to B and C. From C, draw CF perpendicular to CE, intersecting the extension of BE at point F. Connect DF and DE. If CE = CF = 1, DE = √6, which of the following conclusions are correct: ① △CBE ≈ △CDF; ② BF ⊥ DF; ③ The distance from point D to CF is 2; ④ S_{DECF} = √2 + 1. The correct conclusions are ______. (Fill in the serial numbers)

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


As shown in the figure, sector $$A O B$$, with $$O B = \sqrt{2}$$, $$\angle A O B = 90 \circ$$, $$C$$ is any point on arc $$A B$$, draw $$C D \bot O B$$ at point $$D$$, let the incenter of $$\triangle O D C$$ be $$E$$, connect $$O E$$ and $$C E$$, when point $$C$$ moves from point $$B$$ to point $$A$$, the path length traced by the incenter $$E$$ is           .

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


In $$R t \triangle A B C$$, $$D$$ is the midpoint of the hypotenuse, $$E$$ and $$F$$ lie on side $$B C$$, $$D E \bot A F$$, $$\angle A E B = 2 \angle C E D = 2 \angle C A F$$, $$C F = 4$$, then $$B E =$$        ; $$E F =$$          .

Answer: $$1$$     $$\frac{\sqrt{17} - 3}{2}$$


As shown in the figure, in $$\triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C = B C = 5 \text{cm}$$; in $$\triangle C D E$$, $$\angle D C E = 90 \circ$$, $$D C = E C = 3 \text{cm}$$; the lines $$B D$$ and $$A E$$ intersect at point F. Now rotate $$\triangle D C E$$ around point C for one full revolution. During the rotation, $$\angle F =$$        $$\circ$$, and the maximum length of segment $$A F$$ is      $$\text{cm}$$.

Answer: 90     7


As shown in the figure, in quadrilateral $$A B C D$$, $$A B =\text{2} \sqrt{\text{3}}$$, diagonal $$A C =\text{2}$$, $$\angle B A C =\angle A C D =\text{60}°$$, let $$A D = k \cdot B D$$, then the minimum value of $$k$$ is            .

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


As shown in the figure, in quadrilateral ABCD, $$\angle BCD = \text{90}°$$, $$BC = \sqrt{\text{3}} CD$$, $$\angle BAC = \text{60}°$$, if $$AB = \text{5}$$, $$AD = \text{2}$$, then the length of segment AC is      .

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


Given square ABCD, AB=6, DP bisects ∠ADC and DP=$$\sqrt{2}$$, AE=2, connect EP and CP, point F is on EP such that EF:FP=11:3. Connect CF, point M is on EC such that ∠EFM=∠CFP, draw MN⊥CF intersecting CP at point N, then PN=          .

Answer: $$\frac{4 \sqrt{26}}{7}$$/$$\frac{4}{7} \sqrt{26}$$


As shown in the figure, in △ABC, ∠BAC = 60°, its perimeter is 20, ⊙I is the incircle of △ABC, with radius $$\sqrt{3}$$, then the diameter of the circumcircle of △BIC is                   .

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


In Rt△ABC, AB = AC = $$4 \sqrt{2}$$, BO = $$\frac{1}{4}$$AB, point M is a moving point on side BC, rotate segment OM counterclockwise around point O by 90° to ON, connect AN and CN, then the minimum perimeter of △CAN is             .

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


As shown in the figure, in △ABC, ∠BAC = 30°, and AB = AC. P is a point inside △ABC. If the minimum value of AP + BP + CP is 4$$\sqrt{2}$$, then the length of BC is         .

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


As shown in the figure, in square $$ABCD$$, points E and F are two moving points on sides $$AB$$ and $$BC$$, respectively, and the perimeter of square $$ABCD$$ is twice the perimeter of $$\triangle BEF$$. Connect $$DE$$ and $$DF$$, which intersect diagonal $$AC$$ at points M and N, respectively. The following conclusions are given: ① If $$AE = 2$$, $$CF = 3$$, then $$EF = 4$$; ② $$\angle EFN + \angle EMN = 180^\circ$$; ③ If $$AM = 2$$, $$CN = 3$$, then $$MN = 4$$; ④ If $$\frac{MN}{AM} = 2$$, $$BE = 3$$, then $$EF = 4$$. The correct conclusion(s) is/are            .

Answer: ②


As shown in the figure, in $$\Delta ABC$$, $$\angle C = 90^{\circ}$$, $$AC = 8$$, $$BC = 6$$, O is the incenter, and a line passing through point O intersects AC and AB at points D and E, respectively. If $$DE = CD + BE$$, then the length of segment CD is          .

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


As shown in the figure, in $$R t \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$A C > B C$$, three squares $$A B H L$$, $$A C D E$$, $$B C F G$$ are constructed outwardly on the three sides of $$\triangle A B C$$, and $$D F$$ is connected. From point $$C$$, draw a perpendicular $$C J$$ to $$A B$$, with foot at $$J$$, intersecting $$D F$$ and $$L H$$ at points $$I$$ and $$K$$, respectively. If $$C I = 5$$, $$C J = 4$$, then the area of quadrilateral $$A J K L$$ is         .

Answer: 80


As shown in the figure, in quadrilateral ABCD, $$\angle B = \angle C = 90^\circ$$, point E lies on side BC, and AE is connected. If $$\angle BAD - \frac{1}{2} \angle BAE = 45^\circ$$, $$AB = BC = 4CD$$, $$AE = 3$$, then the length of segment AD is      .

Answer: $$\frac{18}{5}$$


As shown in the figure, in an equilateral triangle ABC with side length 2, P0 is the midpoint of side BC. A beam of light emanates from P0, strikes point P1 on AC, then reflects successively to points P2 and P3 on AB and BC, respectively, with 1 < BP3 < $$\frac{3}{2}$$ (angle of reflection equals angle of incidence). The range of values for P1C is     .

Answer: $$1 < P_{1} C < \frac{7}{6}$$


As shown in the figure, a circle ⊙O with radius 2 is tangent to line l at point A. P is a moving point on ⊙O (not coinciding with point A). From point P, draw PB ⊥ l, with foot at B, and connect PA. Let PA = x and PB = y. Then the maximum value of (x - y) is        .

Answer: 1


As shown in the figure, in △ABC, $$\angle A B C = \angle A C B$$, point O is the circumcenter of △ABC. Connect CO and extend it to intersect side AB at point P, $$A P = 3$$, $$B P = 4$$, then the value of $$c o s \angle A B C$$ is        .

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


As shown in the figure, in △ABC, $$\angle BCA = 90^{\circ}$$, $$AC = 6$$, $$BC = 8$$, point D is the midpoint of AB, and point E is a moving point on BC. Triangle △BDE is folded along DE to obtain △FDE, and EF intersects CD at point M. When △DFM is a right triangle, the length of segment BE is   .

Answer: $$\frac{11}{2}$$ or 7


Given: As shown in the figure, in rectangle ABCD, AB = 6, BC = 9. Point E is a point on diagonal AC. The circle ⊙O passing through points C, D, and E intersects AD and BC at points F and G respectively. Connect ED, EF, and EG. Extend GE to intersect AD at point H. If △HEF is an isosceles triangle, then the length of CE is          .

Answer: 6 or $$\frac{3}{2} \sqrt{13}$$ or $$\frac{24 \sqrt{13}}{13}$$


As shown in the figure, in the right triangle $$\triangle$$ABC, ∠ACB=90°, ∠B=30°, AC=2, D is a moving point on BC, EF is the perpendicular bisector of AD, intersecting AC at E and AB at F. Then the maximum value of BF is

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


In parallelogram ABCD, from point C, draw perpendiculars to AB and AD, with feet at F and E respectively. Connect BE and CF, intersecting at point G. Point N lies on CE, and connect FN intersecting BE at point M. Given ∠A = 120°, AE = 2ED, and CN = $$2 \sqrt{3}$$, and FE = FN, then $$S_{\triangle F M G}$$ =

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


As shown in the figure, the side length of square ABCD is 8, and P is a moving point on side CD. EF ⊥ BP intersects BP at G, and EF bisects the area of square ABCD. The minimum value of segment GC is   .

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


As shown in the figure, in parallelogram $$\square A B C D$$, $$\angle C = 60^\circ$$, connect DB, $$D B \bot B C$$, rotate $$\triangle A D B$$ counterclockwise about point A to $$\triangle A D^{'} B^{'}$$, draw $$B^{'} E \parallel D B$$ intersecting line $$D^{'} D$$ at point E, connect $$B B^{'}$$ intersecting $$D^{'} E$$ at point F. If $$B^{'} E = 13 \sqrt{3}$$, $$D F = 10$$, then $$A F =$$      .

Answer: $$12 + 5 \sqrt{3}$$/$$5 \sqrt{3} + 12$$


As shown in the figure, in quadrilateral ABCE, ∠B = ∠A, ∠E = 90°, point D lies on AB, and AD : BD = 5 : 11. Connect CD. If point D lies on the perpendicular bisector of CE and satisfies ∠A = 2∠BDC, and CE = 10, then the length of segment AB is      .

Answer: 20


As shown in the figure, in △ABC, ∠ABC = 45°, AD and BE are the altitudes to sides DC and AC respectively. Connect DE. From point D, draw DF ⊥ DE intersecting BE at point F. G is the midpoint of BE. Connect AF and DG. The relationship between AF and DG is    .

Answer: $$A F = 2 D G$$,$$A F \bot D G$$/$$A F \bot D G$$,$$A F = 2 D G$$


As shown in the figure, in rectangle ABCD, AB = 2, BC = 3, point E is the midpoint of side AB, connect CE, fold triangle BCE along CE to obtain triangle FCE, and let CF intersect BD at point P. Then the length of DP is    .

Answer: $$\frac{8 \sqrt{13}}{17}$$


As shown in the figure, in $$\triangle$$ABC, AB=AC, ∠ABC=30°, D is a moving point on side AB (excluding point B). Construct a square CDEF with CD as one side, and connect BE. Let BD=x, and let the area of $$\triangle$$BDE be s. If s=ax2+6x, where a is a constant, then the length of segment AB is     .

Answer: $$\left(\right. \frac{4}{3} a + \frac{2}{3} \left.\right) x + 8$$.


As shown in the figure, in square ABCD, diagonals AC and BD intersect at point O. F is a moving point on segment OD (point F does not coincide with points O or D). Connect CF, and from point F draw FG ⊥ CF intersecting AC and AB at points H and G, respectively. Connect CG intersecting BD at point M. Draw OE ∥ CD intersecting CG at point E, and let EF intersect AC at point N. The following conclusions are given: ① When BG = BM, AG = √2 BG; ② OH/OM = OF/OC; ③ When GM = HF, CF² = CN · BC; ④ CN² = BM² + DF². Among these, the correct ones are ______ (fill in the serial numbers only).

Answer: ①③④


As shown in the figure, in rhombus ABCD, BC = 9, ∠ABC = 60°, point E lies on BC such that BE = 6. Triangle ABE is folded along AE to obtain triangle AB′E, where B′E intersects CD at point F. Then CF =            .

Answer: $$\frac{9}{5}$$


As shown in the figure, in rhombus ABCD, AB = 2, DE ⊥ BC at point E, F is the midpoint of CD, connect AF and EF. If ∠AFE = 90°, then the length of CE is         .

Answer: $$\sqrt{3} - 1$$;


As shown in the figure, in rectangle ABCD, AB = 5, BC = 8, point M is the midpoint of side BC, and P is a moving point on line AD. Construct right triangle MPQ on the right side of MP such that PM = PQ. Connect AM and AQ. The minimum perimeter of triangle AMQ is   .

Answer: $$\sqrt{221}$$+$$\sqrt{41}$$


As shown in the figure, in square $$ABCD$$, E and F are moving points on ray $$BD$$, and $$\angle EAF = 45^\circ$$. Rays $$AE$$ and $$AF$$ intersect the extensions of $$BC$$ and $$CD$$ at G and H, respectively. Connect $$EC$$. Among the following conclusions: ① $$AE = CE$$; ② $$\triangle AEF \sim \triangle GHC$$; ③ $$BG = GH + DH$$; ④ $$EF^2 = BE^2 + DF^2$$; ⑤ If $$AB = 3CH$$, then $$CD = 2CG$$; ⑥ $$S_{\triangle AGH} : S_{\triangle BCD} = GH : AB$$, the ones that are definitely correct are ______. (Write the correct sequence numbers on the line)

Answer: ①③④⑥


As shown in the figure, in rectangle ABCD, E is a point on side BC. Connect AE. From point B, draw BF perpendicular to AE at point G, intersecting line CD at point F. Construct parallelogram BEHF with BE and BF as adjacent sides. Let M be the midpoint of BH. Connect GM. If AB = 3 and BC = 2, then the minimum value of GM is       .

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


As shown in the figure, in quadrilateral $$A B C D$$, connect diagonals $$A C$$ and $$B D$$. Given $$A B = 6$$, $$A C = C D$$, $$\angle A C B = 45 \circ$$, $$\angle A C D = 90 \circ$$, the maximum value of diagonal $$B D$$ is      .

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


As shown in the figure, the acute vertex A of the isosceles right triangle ABC lies on a circle ⊙O, with ∠ACB = 90°. The legs AC and hypotenuse AB intersect ⊙O at points E and D, respectively. Tangents to ⊙O are drawn through points D and E, intersecting at point F, and point F happens to lie on leg BC. Connect OC, OD, and OE. If the radius of ⊙O is 4, then the maximum value of OC is        .

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


As shown in the figure, quadrilateral ABCD is a square, E is any point on side BC, ∠AEF = 90°, and EF intersects the angle bisector CF of the exterior angle of the square at point F. From point C, draw CH ⊥ DF, intersecting the extension of DF at point H. If AB = 4 and BE = $$\frac{1}{3}$$BC, then CH =     .

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


As shown in the figure, in the equilateral triangle $$ABC$$, D is the midpoint of $$AC$$, and P is a moving point on side $$AB$$. From point P, draw $$PE \bot AB$$, intersecting $$BC$$ at point E. Connect $$DP$$ and $$DE$$. If $$AB = 8$$ and $$\triangle PDE$$ is an isosceles triangle, then the length of $$BP$$ is                 .

Answer: $$- 3 + \sqrt{33}$$ or $$4$$ or $$12 - 4 \sqrt{6}$$.


As shown in the figure, in square ABCD with side length 4, P is a moving point on side BC (excluding points B and C). Triangle ABP is folded along line AP, and point B lands at point E. On side CD, there is a point M such that when triangle CMP is folded along line MP, point C lands at point F on line PE. Line PE intersects CD at point N, and connect MA and NA. When triangle ABP is congruent to triangle ADN, the length of BP is ______.

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


As shown in the figure, in square ABCD, point E lies on ray BC. Connect AB and DE, then the minimum value of $$\frac{D E}{A E}$$ is     .

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


In $$\triangle A B C$$, $$A B = 4 \sqrt{2}$$, $$A C = 6$$, $$\angle A = 45^\circ$$. Fold $$\triangle A B C$$ so that point $$C$$ lands on point $$D$$ on side $$A B$$, with crease $$E F$$ intersecting $$A C$$ at point $$E$$. As point $$D$$ moves continuously from $$B$$ toward $$A$$, the path length traversed by point $$E$$ is denoted as $$m$$. Then $$B C =$$     , $$\text{m} =$$        .

Answer: $$2 \sqrt{5}$$     $$20 - 12 \sqrt{2}$$


As shown in the figure, the area of rhombus $$A B C D$$ is $$8 \sqrt{3}$$, $$\angle B A D = 60 \circ$$, diagonals $$A C$$ and $$B D$$ intersect at point $$O$$. If point $$P$$ is a point on diagonal $$A C$$, then the minimum value of $$\frac{1}{2} A P + B P$$ is       .

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


As shown in the figure, $$\Delta A B C$$, with circumcenter $$O$$, $$B C = 12$$, $$\angle B A C = 60^\circ$$, construct isosceles right triangles $$\Delta A B D$$ and $$\Delta A C E$$ outwardly on sides $$A B$$ and $$A C$$ respectively, connect $$B E$$ and $$C D$$ intersecting at point $$P$$, then the minimum value of $$O P$$ is      .

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


Given: As shown in the figure, isosceles right triangle $$\triangle A B C$$, $$\angle B A C = 90 \circ$$, $$A B = A C$$, point D is a point outside $$\triangle A B C$$, $$\angle A D B = 45 \circ$$, connect CD, $$A D = 4$$, $$C D = 5 \sqrt{2}$$, the length of BC is        .

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


As shown in the figure, in rectangle $$A B C D$$, $$A D = 5, A B = 3$$, E is a moving point on $$B C$$, connect $$A E$$, construct $$D F \bot A E$$ at F, connect $$C F$$, when $$\triangle C D F$$ is an isosceles triangle, then the length of $$B E$$ is          .

Answer: $$\frac{5}{2}$$ or 4 or 1.


As shown in the figure, in quadrilateral $$A B C D$$, $$A C \bot B C$$ at point $$C$$, $$\angle B A C = \angle A D C$$, and $$B C = \frac{3}{4} A C$$. When $$C D = 4$$ and $$A D = 2$$, the length of segment $$B D$$ is      .

Answer: $$\frac{\sqrt{109}}{2}$$


As shown in the figure, in square $$ABCD$$, $$AB = 9$$, point $$E$$ is a point on $$AD$$ such that $$AE : ED = 1 : 2$$, point $$P$$ is a moving point on side $$AB$$, connect $$PE$$, draw $$EF \bot PE$$ intersecting ray $$BC$$ at point $$F$$, connect $$PF$$, point $$M$$ is the midpoint of $$PF$$, connect $$DM$$, then the minimum value of $$DM$$ is        .

Answer: $$\frac{21 \sqrt{10}}{10}$$


As shown in the figure, in quadrilateral ABCD, AB = AD = CD = 4, AD ∥ BC, ∠B = 60°, points E and F are two moving points on sides BC and CD respectively, and ∠EAF = 60°. Then the minimum area of △AEF is     .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = 4$$, $$A C = 5$$, an equilateral triangle $$\triangle B C D$$ is constructed outwardly on side $$B C$$, and $$A D$$ is connected. Then the maximum value of $$A D$$ is          .

Answer: 9


As shown in the figure, in the equilateral triangle $$\Delta A B C$$, $$A B = 3$$, points $$D$$ and $$E$$ are moving points on sides $$B C$$ and $$C A$$ respectively, with $$B D = C E$$. Connect $$A D$$ and $$B E$$, intersecting at point $$F$$. When point $$D$$ moves from point $$B$$ to point $$C$$, the length of the path traced by point $$F$$ is         .

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


As shown in the figure, in △ABC, AB = BC = 5, AD ⊥ BC at point D, and AD = $$3$$. If point M is a moving point on AD, and point P starts from point B, moving along BM at a speed of $$1$$ unit per second, then along MA at a speed of $$\sqrt{10}$$ units per second, the shortest time for point P to complete its motion is    seconds.

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


As shown in the figure, in $$\triangle A B C$$, $$A B = A C = 2, \angle B A C = 120 \circ$$, P is a moving point on side $$B C$$, connect $$A P$$, rotate segment $$A P$$ clockwise around point A by $$120 \circ$$ to obtain $$A P^{'}$$, then the minimum value of segment $$P P^{'}$$ is       .

Answer: $$\sqrt{3}$$


As shown in the figure, in △ABC, AB = AC, ∠ABC = α, tanα = $$\frac{5}{6}$$, AD ⊥ BC at point D, point E is a moving point on segment AD, connect EB, rotate segment EB counterclockwise by 2α about point E to obtain segment EF, connect AF. If BC = 24, then the minimum value of segment AF is       .

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


As shown in the figure, in △ABC, ∠C = 90°, AC = 8, BC = 16, point D is on side BC. Fold △ABC along DE such that point B coincides with point A. Connect AD. Point P lies on segment AD. When the distance from point P to the right-angle sides of △ABC is 5, the length of AP is     .

Answer: $$\frac{25}{3}$$ or $$\frac{15}{4}$$.


As shown in the figure, in square ABCD, O is the midpoint of BD. An equilateral triangle BCE is constructed inward on side BC. Connect AE and extend it to intersect CD at F. Connect BD, intersecting CE and AF at G and H, respectively. The following conclusions: ① $$\angle C E H = 45 \circ$$; ② $$B G = \sqrt{2} D G$$; ③ $$G F // E D$$; ④ $$2 O H + D H = B D$$; ⑤ $$S_{\triangle B E C} : S_{\triangle B G C} = \frac{\sqrt{3} + 1}{2}$$, among which the correct ones are          .

Answer: ①③⑤


As shown in the figure, in $$\Delta A B C$$, $$A B = A C$$, $$B C = 12$$, $$D$$ is the midpoint of side $$A C$$, the perpendicular bisector of segment $$B D$$ intersects side $$B C$$ and $$A B$$ at points $$E$$ and $$F$$, respectively. Connect $$D F$$ and $$E F$$. Let $$B E = x$$, $$t a n \angle A C B = y$$. The following conclusions are given: ① $$D F / / B C$$; ② The area of $$\Delta B D E$$ is $$\frac{3}{2} x y$$; ③ The perimeter of $$\Delta C D E$$ is $$12 + x$$; ④ $$x^{2} - y^{2} = 9$$; ⑤ $$2 x - y^{2} = 9$$. Among these, the correct conclusions are       (fill in the serial numbers of all correct conclusions).

Answer: ②⑤


As shown in the figure, in square ABCD, AB = 12, points E, F, G, H lie on AB, BC, CD, DA respectively, EG = 13, FH = 15, then the area of quadrilateral EFGH is    .

Answer: 94.5


As shown in the figure, given $$\square A B C D$$, $$B E \bot A D$$ at $$E$$, $$F$$ is the midpoint of $$A B$$, connect $$E F$$, translate $$\Delta B E F$$ to the right to $$\Delta H D G$$, such that $$B$$ coincides with $$H$$, $$E$$ coincides with $$D$$, and $$F$$ coincides with $$G$$, connect $$D F$$, $$H F$$, $$E H$$, if $$G$$ is the orthocenter of $$\Delta D F H$$, $$A E = 2$$, $$B H = 8$$, then the distance from $$F$$ to $$E H$$ is        .

Answer: 3


As shown in the figure, in rectangle ABCD, E is a point on CD. If triangle ADE is folded along line AE such that point D lands at point D' on side BC. F is a point on AD such that DF = CD'. EF intersects BD at point G, and AD' intersects BD at point H. D'E is parallel to BD, and HG = 4. Then BD =  .

Answer: 6+2$$\sqrt{5}$$.


As shown in the figure, in rectangle $$A B C D$$, $$A B = 4, A D = 6$$, $$E$$ and $$F$$ are the midpoints of $$A D$$ and $$B C$$ respectively, $$G$$ and $$H$$ lie on $$D C$$ and $$A B$$ respectively, and $$\angle B E G = \angle D F H = 90 \circ$$. Connect $$B G$$ and $$D H$$. The perimeter of the hexagon $$I J K L M N$$, which is the overlapping region of $$\triangle B E G$$ and $$\triangle D F H$$, is

Answer: 9.8


As shown in the figure, square paper ABCD is folded along line BE, and point C lands at point G. Connect BG and extend it to intersect CD at point H. Extend EG to intersect AD at point F, and connect FH. If AF = FD = 6 cm, then the length of FH is     cm.

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


As shown in the figure, in square $ABCD$ with side length $4$, point $E$ is the quarter-point closer to point $D$, and point $F$ is the midpoint of $AB$. Fold $\triangle AEF$ along $EF$ to obtain $\triangle A'EF$, and connect $A'C$. Then the distance from point $D$ to the line containing $A'C$ is                .

Answer: $$\frac{28}{65} \sqrt{65}$$


As shown in the figure, in square $$ABCD$$, $$\triangle BPC$$ is an equilateral triangle, and the extensions of $$BP$$ and $$CP$$ intersect $$AD$$ at points $$E$$ and $$F$$, respectively. Connect $$BD$$ and $$DP$$, and let $$BD$$ intersect $$CF$$ at point $$H$$. The following conclusions are given:
① $$\triangle ABE \cong \triangle DCF$$; ② $$\frac{FP}{PH} = \frac{\sqrt{3}}{3}$$; ③ $$DP^{2} = PH \cdot PB$$; ④ $$\frac{S_{\triangle HPD}}{S_{\text{square } ABCD}} = \frac{\sqrt{3} - 1}{2}$$, among which the correct ones are      . (Write all correct conclusion numbers)

Answer: ①②③.


As shown in the figure, $$C$$ is a moving point on segment $$A E$$ (not coinciding with points $$A$$ and $$E$$). On the same side of $$A E$$, construct equilateral triangle $$A B C$$ and equilateral triangle $$C D E$$. Let $$A D$$ intersect $$B E$$ at point $$O$$, $$A D$$ intersect $$B C$$ at point $$P$$, and $$B E$$ intersect $$C D$$ at point $$Q$$. Connect $$P Q$$. The following conclusions: ① $$A D = B E$$; ② $$P Q / / A E$$; ③ $$\angle A O B = 60 \circ$$; ④ $$\triangle C P Q$$ is an equilateral triangle. The ones that always hold are      .

Answer: ①②③④


As shown in the figure, in $$\Delta A B C$$, $$A B = 10$$, $$A C = 8$$, $$B C = 6$$, a moving circle passing through point $$C$$ and tangent to side $$A B$$ intersects $$C A$$ and $$C B$$ at points $$P$$ and $$Q$$, respectively. The minimum length of segment $$P Q$$ is          .

Answer: $$4.8$$


As shown in the figure, the fixed line $$l$$ passes through the center $$O$$, and $$P$$ is a moving point on the radius $$OA$$. $$AC \bot l$$ at point $$C$$. When the radius $$OA$$ rotates around point $$O$$, it always holds that $$OP = OC$$. When $$OA$$ rotates by $$60^\circ$$ around point $$O$$, the ratio of the path lengths traveled by points $$P$$ and $$A$$ is .

Answer: 1.


As shown in the figure, given rectangle paper $$ABCD$$, $$AD = 15$$, $$AB = 8$$, point $$E$$ is on side $$BC$$. Fold the paper along $$AE$$ such that point $$B$$ lands at point $$F$$. Connect $$FC$$. When $$\Delta EFC$$ is a right triangle, the area of $$\Delta EFC$$ is       .

Answer: 28 or $$\frac{108}{5}$$


As shown in the figure, in right triangle ABC, ∠ABC = 90°, point D is the midpoint of AC. Draw the angle bisector DE of ∠ADB, intersecting AB at point E. AE = 6, DE = 10. Point P lies on side BC, and triangle DEP is an isosceles triangle. Find the length of BP:

Answer: 2 or 5 or 8 or 18


In square ABCD, point G is on AB, point H is on BC, and ∠GDH = 45°. DG and DH intersect diagonal AC at points E and F, respectively. The quantitative relationship among segments AE, EF, and FC is _______.

Answer: $$E F^{2} = A E^{2} + C F^{2}$$


As shown in the figure, in square ABCD, AB = 2. Extend BC to point E such that CE = 1, and connect DE. A moving point P starts from point A and moves along AB-BC-CD-DA toward point A at a speed of 1 unit per second. Let the motion time of point P be t seconds. When triangles ABP and DCE are congruent, the value of t is    .

Answer: 3 or 7


Given that the height of the equilateral triangle △ABC is 6, there is a point P in the plane of this triangle. If the distance from point P to line AB is 1, and the distance from point P to line AC is 3, then the distance from point P to line BC may be       .

Answer: 2, 4, 8, 10


As shown in the figure, $$BD$$ is the diagonal of quadrilateral $$ABCD$$, $$BC = AD$$, $$\angle A = \angle CBD$$, $$\angle ABD = 120^\circ$$, $$AB = 3$$, $$CD = \sqrt{19}$$, then the length of $$BC$$ is             .

Answer: 7


As shown in the figure, in parallelogram ABCD, ∠A = 60°, points E and F lie on sides AD and DC respectively, with DE = DF, and ∠EBF = 60°. If AE = 2 and FC = 3, then the length of EF is                 .

Answer: $$\sqrt{21}$$


As shown in the figure, AB is the diameter of a semicircle, and C is a point on the semicircular arc. One side DG of square DEFG lies on the diameter AB, and the other side DE passes through the incenter O of triangle ABC, with point E lying on the semicircular arc.

① If vertex F of the square also lies on the semicircular arc, then the ratio of the radius of the semicircle to the side length of the square is ______;

② If the area of square DEFG is 100, and the inradius r of triangle ABC is 4, then the diameter AB of the semicircle is ______.

Answer: $$\sqrt{5}$$:2     21


As shown in the figure, points E and F are moving points on sides BC and CD of square ABCD with side length 2, and BE = CF. Connect DE and AF, intersecting at point P. From point P, draw PN perpendicular to CD at point N and PM perpendicular to BC at point M. Connect MN; find the minimum length of MN.

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


As shown in the figure, in quadrilateral ABCD, diagonals AC and BD intersect at point E. If AC bisects ∠DAB, and AB = AD, CD = CB, the following four conclusions hold: ① AC ⊥ BD; ② BE = DE; ③ ∠DAB = 2∠BAC; ④ △ABD is an equilateral triangle. Please write the serial numbers of the correct conclusions

Answer: ①②③


As shown in the figure, in right triangle ABC, AC = BC, ∠ACB = 90°, point D is a point on side AB. CD is rotated 90° clockwise about point D to DE, and CE intersects AB at point G. Given that AD = 8, BG = 6, point F is the midpoint of AE. Connect DF. Find the length of segment DF.

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


As shown in the figure, in parallelogram $$A B C D$$, $$D E$$ bisects $$\angle A D C$$, $$A D = 6$$, $$B E = 2$$, then the perimeter of parallelogram $$A B C D$$ is      .

Answer: 20


As shown in the figure, points B, F, C, E lie on a straight line, $$BF = CE$$, $$\angle B = \angle E$$, add one condition to make $$\triangle ABC \cong \triangle DEF$$, this added condition can be      (only one is required, no auxiliary lines needed).

Answer: $$A B = D E$$ or $$∠A=∠D$$ or $$∠ACB=∠EFD$$ or $$AC∥DF$$, the answer is not unique.


As shown in the figure, $$AD$$ is the median of $$\triangle ABC$$, $$BE$$ is the median of $$\triangle ABD$$, $$EF \bot BC$$ at point $$F$$. If $$S_{\triangle ACD} = 12 \text{cm}^{2}$$, $$BD = 4 \text{cm}$$, then the length of $$EF$$ is        $$\text{cm}$$.

Answer: 3


As shown in the figure, $$MN$$ is a chord of $$\bigodot O$$, $$\angle N = 50^\circ$$, then the measure of $$\angle MON$$ is      .

Answer: $$80 \circ$$


As shown in the figure, $$AB$$ is the diameter of ⊙$$O$$, $$AC$$ is the tangent to ⊙$$O$$, with $$A$$ as the point of tangency. Connect $$BC$$, which intersects ⊙$$O$$ at point $$D$$, and connect $$OD$$. If $$\angle AOD = 82^\circ$$, then $$\angle C =$$         $$^\circ$$.

Answer: 49


As shown in the figure, in $$\triangle A B C$$, the perpendicular bisector of $$A C$$ intersects $$B C$$ at point D and $$A C$$ at point E. Connect $$A D$$. If the perimeter of $$\triangle A B C$$ is 13 and $$A E = 2$$, then the perimeter of $$\triangle A B D$$ is    .

Answer: 9


As shown in the figure, in $$\triangle A B C$$, $$A B = 10$$, $$B C = 14$$, points D and E are the midpoints of sides $$A B$$ and $$A C$$ respectively, point F is a point on segment $$D E$$, connect $$A F$$ and $$B F$$, if $$\angle A F B = 90^\circ$$, then the length of segment $$E F$$ is        .

Answer: 2


As shown in the figure, in parallelogram $$\square A B C D$$, $$A B = 4$$, $$B C = 6$$, the perpendicular bisector of $$A C$$ intersects $$A D$$ at point $$E$$, and connect $$C E$$. The perimeter of $$\triangle C D E$$ is       .

Answer: 10


As shown in the figure, the side lengths of $$\triangle A B C$$ are $$A B$$, $$B C$$, $$C A$$, which are 10, 15, 20 respectively. Its three angle bisectors intersect at point O. Connect OA, OB, OC, dividing $$\triangle A B C$$ into three triangles. Then $$S_{\triangle A B O} : S_{\triangle B C O} : S_{\triangle C A O}$$ equals          .

Answer: 2:3:4


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A = 90 \circ$$, point D lies on side AB, and CD is connected. If $$B D = C D$$ and $$\frac{A D}{B D} = \frac{1}{3}$$, then $$t a n B =$$     .

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


As shown in the figure, in △ABC, AC = 2, BC = 4, point O lies on BC, and a circle with radius OB is tangent to AC at point A. D is a moving point on side BC. When △ACD is a right triangle, the length of AD is           .

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


As shown in the figure, the radius of $$\bigodot O$$ is $$2$$ $$\text{cm}$$, $$AB$$ is a chord of $$\bigodot O$$, and point $$C$$ is a point on $$\overset{⌢}{AB}$$. When $$\overset{⌢}{AB}$$ is folded along chord $$AB$$, point $$C$$ coincides with the center $$O$$. The area of the shaded region is       . (Express the result in terms of $$\pi$$ and radicals)

Answer: $$\left(\frac{2}{3} \pi - \sqrt{3}\right) \text{cm}^{2}$$


As shown in the figure, rectangle paper $$ABCD$$ is folded along $$EF$$, and after folding, point $$A$$ lands at $$A^{'}$$, and point $$B$$ coincides exactly with point $$D$$. Given that $$\angle DFC = 60^\circ$$, $$CF = 3$$, the length of $$AE$$ is      .

Answer: $$3$$


As shown in the figure, in $$\triangle A B C$$, $$\angle C = 90 \circ$$, $$A D$$ is the angle bisector of $$\triangle A B C$$, $$D E \bot A B$$ at point E, $$C D = 3$$, $$A B = 10$$, then the area of $$\triangle A B D$$ is          .

Answer: 15


As shown in the figure, the left and right sides of the alley are vertical walls. When the ladder leans against the left wall, the distance from the bottom of the ladder to the left wall corner is $$0.7$$ meters, and the top of the ladder is 2.4 meters above the ground. If the position of the bottom of the ladder remains unchanged and the ladder leans against the right wall, the top of the ladder is 2 meters above the ground. The width of the alley is      meters.

Answer: $$2.2$$/$$2 \frac{1}{5}$$/$$\frac{11}{5}$$


As shown in the figure, in the right triangle $$\text{Rt} \triangle A B C$$, squares are constructed outwardly on each of its three sides, with areas denoted as $$S_{1}, S_{2}, S_{3}$$ respectively. If $$S_{3} + S_{2} - S_{1} = 18$$, then the area of the shaded region in the figure is

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


As shown in the figure, in $$\triangle A B C$$, $$A B = 8 \text{cm}$$, $$B C = 6 \text{cm}$$. With point $$B$$ as the center, draw an arc with any length as the radius, intersecting sides $$A B$$ and $$B C$$ at points $$D$$ and $$E$$, respectively; then, with $$D$$ and $$E$$ as centers, draw arcs with radii greater than $$\frac{1}{2} D E$$, intersecting at point $$F$$; draw ray $$B F$$ intersecting side $$A C$$ at point $$G$$. If the area of $$\triangle A B G$$ is $$20 \text{cm}^{2}$$, then the area of $$\triangle C B G$$ is      $$\text{cm}^{2}$$.

Answer: 15


As shown in the figure, in $$\triangle A B C$$, point D is a moving point on $$A B$$, connect $$C D$$, point E is a point on segment $$C D$$, points F and G are moving points on sides $$C A$$ and $$C B$$ respectively. If $$\angle A C B = 30 \circ$$, $$C E = 6$$, then the minimum perimeter of $$\triangle E F G$$ is     .

Answer: 6


As shown in the figure, arrange a set of triangle boards as shown in the figure, in which $$\angle 1 =$$        .

Answer: $$120^{\circ}$$


As shown in the figure, in $$\triangle A B C$$, $$B E$$ and $$C E$$ are the angle bisectors of $$\angle A B C$$ and $$\angle A C B$$, respectively, $$E D \parallel A C$$ intersects $$B C$$ at point D, and $$E F \bot A B$$ at point F. If $$B C = 35$$, $$E F = 5$$, $$D E = 13$$, then the area of $$\triangle E B D$$ is        .

Answer: $$55$$


As shown in the figure, which is a pattern formed by two set squares, the measure of $$\angle A E D$$ is      $$\circ$$.

Answer: $$105$$


As shown in the figure, in $$\triangle A B C$$, $$\angle C = 90 \circ$$, $$C A = C B$$, $$A D$$ bisects $$\angle C A B$$ and intersects $$B C$$ at D, $$D E \bot A B$$ at E. If $$A B = 6$$, the perimeter of $$\triangle D E B$$ is       .

Answer: 6


As shown in the figure, points $$B$$, $$E$$, $$C$$, $$F$$ lie on a straight line, $$AC \parallel DF$$, $$BE = CF$$. Only one additional condition is needed to prove $$\triangle ABC \approx \triangle DEF$$; this condition can be        (write one such condition).

Answer: $$A C = D F$$ or $$\angle A = \angle D$$ or $$\angle A B C = \angle D E F$$ or $$A B \parallel D E$$ (The answer is not unique).


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, construct squares $$A C D E$$, $$C B G F$$, and $$A H I B$$ outwardly on the three sides of $$\triangle A B C$$. Let P be a point on $$H I$$. Denote the areas of squares $$A C D E$$ and $$A H I B$$ as $$S_{1}$$ and $$S_{2}$$, respectively. If $$S_{1} = 16$$ and $$S_{2} = 25$$, then the area of quadrilateral $$A C B P$$ is          .

Answer: 18.5


Place a set of triangle rulers as shown in the figure; then $$\angle A O B$$ equals   .

Answer: $$105 \circ$$


As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C, B D$$ intersect at point $$O$$, $$A C = 16$$, $$B D = 12$$, point $$E$$ is the midpoint of $$C D$$, connect $$O E$$, then the length of $$O E$$ is        .

Answer: 5


As shown in the figure, with endpoints $$A$$ and $$B$$ of segment $$AB$$ as centers, arcs are drawn with radius greater than $$\frac{1}{2} AB$$, intersecting at points $$M$$ and $$N$$. Draw line $$MN$$, and let point $$C$$ be a point on line $$MN$$. Connect $$CB$$ and $$CA$$. With $$C$$ as center and $$CB$$ as radius, draw an arc intersecting the extension of $$AC$$ at point $$D$$. Connect $$BD$$. If $$\angle BDC = 25^\circ$$, then the measure of $$\angle BAC$$ is             .

Answer: 65°


As shown in the figure, in rectangle $$A B C D$$, diagonals $$A C$$, $$B D$$ intersect at point $$O$$, $$A C = 8$$, $$A E \bot B D$$, with foot at $$E$$, if $$\angle A B D = 2 \angle C B D$$, then the length of $$B E$$ is       .

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


Given: As shown in the figure, in $$\triangle A B C$$, $$B O, C O$$ are the angle bisectors of $$\angle A B C$$ and $$\angle A C B$$, respectively. A line passing through point O intersects $$A B$$ and $$A C$$ at points D and E, respectively, and $$D E / / B C$$. If $$A B = 6 \text{cm}, A C = 8 \text{cm}$$, then the perimeter of $$\triangle A D E$$ is      .

Answer: $$14 \text{cm}$$


As shown in the figure, in parallelogram $$\square A B C D$$, $$B A = B D$$. With point $$A$$ as the center and $$A D$$ as the radius, draw an arc intersecting $$B D$$ at another point $$F$$. Then, with points $$D$$ and $$F$$ as centers, draw arcs with radii greater than $$\frac{1}{2} D F$$, intersecting at point $$G$$. Draw ray $$A G$$ intersecting $$B D$$ at point $$E$$. If $$\angle C = 70^\circ$$, then the measure of $$\angle D A E$$ is $$\circ$$.

Answer: 20


As shown in the figure, in $$\triangle A B C$$, points D, E, F are the midpoints of sides $$B C$$, $$A D$$, and $$C E$$ respectively, and $$S_{\triangle A B C} = 16$$. Then the area of the shaded region $$S_{shade} =$$       .

Answer: 4


As shown in the figure, in $$\triangle A B C$$, if $$A C = A D = D B$$, and $$\angle B = 24^\circ$$, then $$\angle B A C =$$        .

Answer: $$108 \circ$$


As shown in the figure, in $$\text{Rt} \triangle A B C$$, $$\angle A C B = 90 \circ$$, $$C D$$ is the altitude, if $$\angle B = 58 \circ$$, then the measure of $$\angle A C D$$ is      .

Answer: $$58 \circ$$


As shown in the figure, FE ∥ ON, OE bisects ∠MON, ∠FEO = 28°, then ∠MFE =  degrees.

Answer: 56


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

Answer: $$36$$


As shown in the figure, an equilateral triangle ABC with side length $$5 \text{cm}$$ is translated 1 cm to the right to obtain the equilateral triangle A'B'C'. At this time, the perimeter of the shaded region is      $$\text{cm}$$.

Answer: $$12$$


In the isosceles right triangle $$\triangle A B C$$, $$A C = B C = \sqrt{3}, \angle A C B = 90^{\circ}$$, D is a point on $$A B$$ such that $$A D = 2$$; an arc is drawn with center $$A$$ and radius $$A D$$, intersecting $$B C$$ at point $$F$$ and the extension of $$A C$$ at point $$E$$. The area of the shaded region in the figure is               .

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


As shown in the figure, the square $$ABCD$$ has side length 4, and point $$E$$ is a point on side $$AB$$ (point $$E$$ coincides with point $$A$$ or $$B$$). From point $$A$$, draw $$AF \bot DE$$, with foot at $$G$$, and $$AF$$ intersects side $$BC$$ at point $$F$$. Connect $$DF$$ and $$EF$$. If the area of $$\triangle DEF$$ is $$\frac{13}{2}$$, then the length of $$AE$$ is        .

Answer: 3 or 1


As shown in the figure, a sector with a central angle of $$90 \circ$$ is cut out from a circular patch with a radius of 2. The area of this sector is        .

Answer: $$2 \pi$$


As shown in the figure, in $$\triangle A B C$$, $$A C = 3$$, $$B C = 4$$, an arc is drawn with point A as the center and $$A C$$ as the radius, intersecting $$A B$$ at point D, with $$B D = 2$$. Then $$\angle A C B =$$        °.

Answer: 90


As shown in the figure, quadrilateral $$A B C D$$ is a rectangle, and diagonals $$A C$$, $$B D$$ intersect at point $$O$$. If $$\angle A O B = 50^{\circ}$$, then $$\angle O C D =$$           .

Answer: $$65 \circ$$


As shown in the figure, $$AB$$ and $$AC$$ are tangents to $$\bigodot O$$, with B and C as the points of tangency. Connect $$OC$$ and extend it to D such that $$CD = OC$$. Connect $$AD$$. If $$\angle BAD = 75^\circ$$, then the measure of $$\angle AOC$$ is      .

Answer: $$65 \circ$$


As shown in the figure, in right triangle $$\text{R} \triangle A B C$$, $$\angle C = 90^\circ$$, $$\angle B = 60^\circ$$, point $$D$$ lies on $$B C$$, $$B D = 4$$, points $$P$$ and $$E$$ are moving points on $$A C$$ and $$A B$$ respectively. When the value of $$D P + E P$$ is minimized, $$B E = 5$$. Find the length of $$A B$$.

Answer: $$14$$


As shown in the figure, $$M$$ and $$N$$ are the midpoints of sides $$AC$$ and $$BC$$ of $$\triangle ABC$$, respectively, and $$AN$$ intersects $$BM$$ at point $$O$$. Then $$\frac{\text{area of } \triangle BON}{\text{area of } \triangle ABC} =$$      .

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


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle A = 50 \circ$$, points D, E, F lie on sides $$B C$$, $$A B$$, $$A C$$ respectively. If $$B D = C F$$, $$B E = C D$$, then $$\angle E D F =$$      degrees.

Answer: $$65 \circ$$


As shown in the figure, in $$\triangle A B C$$ and $$\triangle D E F$$, points B, F, C, E lie on the same straight line, $$A B = D E$$, $$\angle B = \angle E$$. Add one condition such that $$\triangle A B C ≌ \triangle D E F$$. The added condition can be      (only one is required, no auxiliary lines needed);

Answer: $$\angle A = \angle D$$ (The answer is not unique)


As shown in the figure, $$AD$$ is the angle bisector of $$\triangle ABC$$, $$DE \bot AB$$ at point $$E$$, if $$AC = 6$$, $$DE = 2$$, then the area of $$\triangle ACD$$ is      .

Answer: 6


In equilateral triangle △ABC, P is a point on BC, and D is a point on AC, such that ∠APD = 60°, BP = 4, CD = 2. Then the side length of △ABC is    .

Answer: 8


As shown in the figure, equilateral triangle $$A B C$$, $$P$$ is a point on $$B C$$, and $$\angle 1 = \angle 2$$, then the measure of $$\angle 3$$ is          (degrees).

Answer: $$60$$


As shown in the figure, the equilateral triangle $$\triangle A B C$$ has side length 3, point $$P$$ is a point on side $$B C$$ such that $$B P = 1$$, and point $$D$$ is a point on side $$A C$$. If $$\angle A P D = 60^\circ$$, then the length of $$C D$$ is        .

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


As shown in the figure, in rectangle $$A B C D$$, $$A B = 6$$, $$B C = 8$$, $$B E \bot A C$$, then $$B E =$$      .

Answer: 4.8


As shown in the figure, in △ABC, AB = BC, AC = 2 cm, $$S_{\triangle A B C} = 3 \text{cm}^{2}$$, the perpendicular bisector of side BC is l, point D is the midpoint of side AC, and point P is a moving point on l. Then the minimum perimeter of △PCD is      .

Answer: 4


As shown in the figure, it is given that $$BO$$ bisects $$\angle CBA$$, $$CO$$ bisects $$\angle ACB$$, and $$MN \parallel BC$$. Let $$AB = 12$$, $$BC = 24$$, $$AC = 18$$; then the perimeter of $$\triangle AMN$$ is       .

Answer: 30


As shown in the figure, in $$\triangle A B C$$, $$A B = A C$$, $$\angle A = 36 \circ$$, $$B D$$ bisects $$\angle A B C$$ and intersects $$A C$$ at point D, point E is the midpoint of $$A B$$, and connect $$D E$$. Then the measure of $$\angle A D B$$ is      .

Answer: $$108 \circ$$


As shown in the figure, point E lies on the side AD of rectangle $$A B C D$$, and $$B C = E C = 8$$, $$\angle A B E = 15^\circ$$. Then the length of $$A B$$ is          .

Answer: 4


As shown in the figure, in the rhombus $$A B C D$$, the diagonals $$A C$$, $$B D$$ intersect at point O, $$B D = 8$$, $$O E \parallel A B$$, intersecting $$B C$$ at point E, $$O E = 2.5$$, then the length of $$A C$$ is       .

Answer: 6


As shown in the figure, $$\triangle A B C$$ is an inscribed triangle of $$\bigodot O$$. If $$\angle O B C = 28^\circ$$, then $$\angle A =$$      .

Answer: $$62 \circ$$


In his work "Ping Sanchao Ju Yao", the early Qing mathematician Mei Wending provided a complete proof of Qin Jiushao's "Sanxie Qiuji Shu", a method for calculating the area of a triangle proposed by the Southern Song mathematician Qin Jiushao. In the proof, he creatively designed a right triangle and derived the conclusion: as shown in the figure, $$AD$$ is the altitude of the acute triangle $$\triangle ABC$$, then $$BD = \frac{1}{2} \left(BC + \frac{AB^{2} - AC^{2}}{BC}\right)$$. When $$AB = 7$$, $$BC = 6$$, and $$AC = 5$$, $$CD =$$    .

Answer: $$1$$


As shown in the figure, quadrilateral $$A B C D$$ is a square, $$\triangle A D E$$ is an equilateral triangle, $$E F \bot A B$$ at point F. If $$A D = 4$$, then $$E F =$$      .

Answer: 2


As shown in the figure, in parallelogram $$\square A B C D$$, $$O$$ is the midpoint of $$B D$$, and line $$E F$$ passes through point $$O$$ and intersects $$A B$$ and $$C D$$ at points $$E$$ and $$F$$, respectively. If $$A E = 10$$, then the length of $$C F$$ is           .

Answer: 10


As shown in the figure, in right triangle ABC, ∠ACB = 90°, $$AC = BC = 2 \sqrt{2}$$, point D is the midpoint of AB, and point P lies on AC such that CP = 1. Rotate CP around point C in the plane; the corresponding point of P is point Q. Connect AQ and DQ. When ∠ADQ = 90°, the length of AQ is      .

Answer: $$\sqrt{5}$$ or $$\sqrt{13}$$/$$\sqrt{13}$$ or $$\sqrt{5}$$


As shown in the figure, in $$\triangle A B C$$, D is a point on $$A B$$, $$C F \parallel A B$$, and points D, E, F are collinear. Please add one condition   , such that $$A E = C E$$. (Only one case needs to be added.)

Answer: $$D E = E F$$ or $$A D = C F$$ (The answer is not unique.)


As shown in the figure, in $$\triangle A B C$$, if $$D E \parallel B C, F G \parallel A C, \angle B D E = 120 \circ, \angle D F G = 115 \circ$$, then $$\angle C =$$        °.

Answer: $$5 5 \circ$$


As shown in the figure, a rectangular paper $$ABCD$$ is folded along crease $$EF$$. After folding, the image of side $$EC$$, denoted as $$EH$$, passes through point $$A$$, and the image of side $$CD$$, denoted as $$HG$$, intersects the extension of $$BA$$ at point $$P$$. If $$PA = PG$$, $$AH = BE$$, and $$CD = 3$$, then the length of $$BC$$ is         .

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


As shown in the figure, in the rhombus $$A B C D$$, $$\angle D A B = 40 \circ$$, connect $$A C$$, and with point $$A$$ as the center and $$A C$$ as the radius, draw an arc intersecting line $$A D$$ at point $$E$$, connect $$C E$$, then the measure of $$\angle A E C$$ is        .

Answer: $$10 \circ$$ or $$80 \circ$$


As shown in the figure, in the equilateral triangle $$\triangle A B C$$, $$A B = 2$$, $$B D$$ is the altitude from $$B$$ to side $$A C$$, and $$B C$$ is extended to point $$E$$ such that $$C E = C D$$. Then the length of $$B E$$ is            .

Answer: 3


As shown in the figure, $$AD$$ is the altitude of the equilateral $$\triangle ABC$$, and M, N are moving points on segments $$AD$$ and $$AC$$ respectively, with $$AM = BN$$. When $$BM + CN$$ is minimized, $$\angle ANC =$$            .

Answer: $$105 \circ$$