Let p be a point in a square ABCD, △ ABP is congruent with △ CBE. If BP = a, let PE be the area of the square with side length

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• 1. As shown in the figure, P is a point in the square ABCD, and there is a point e outside the square ABCD, which satisfies ∠ Abe = ∠ CBP, be = BP,
• 2. It is known that, as shown in the figure, P is a point inside the square ABCD, and there is a point e outside the square ABCD, satisfying that the angle Abe = angle CBP, be = BP, If PA: Pb = 1:2, angle APB = 135 degrees, what is the chord value of angle PAE?
• 3. In square ABCD, point P is a point on edge CD, and PE ⊥ dB, PF ⊥ Ca, perpendicular feet are points E and f respectively, What is the quantitative relationship between PE + PF and the diagonal length of a square
• 4. As shown in the figure, P is a moving point on the diagonal AC of the square ABCD with side length 1 (P does not coincide with A. C), and point E is on the ray BC And PE = PB Verification (1) PE vertical PD Let AP = x △ PBE area be y Find out the function relation between Y and X, and write out the value range of X
• 5. As shown in the figure, P is a point on the diagonal AC of ABCD with side length 1 (P does not coincide with a and C), and point E is on the ray BC, and PE = Pb. (1) verification: PE = PD; (2) PE ⊥ PD
• 6. As shown in Figure 1, it is known that P is a point on the diagonal AC of square ABCD (not coincident with a and C), PE ⊥ BC at point E, PF ⊥ CD at point F. (1) verification: BP = DP; (2) as shown in Figure 2, if quadrilateral PECF rotates counterclockwise around point C, is there always BP = DP in the process of rotation? If so, please prove it; if not, please explain it with counter examples; (3) try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF respectively, so that the length of the two segments is always equal when quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion
• 7. As shown in Figure 1, it is known that P is a point on the diagonal AC of square ABCD (not coincident with a and C), PE ⊥ BC at point E, PF ⊥ CD at point F. (1) verification: BP = DP; (2) as shown in Figure 2, if quadrilateral PECF rotates counterclockwise around point C, is there always BP = DP in the process of rotation? If so, please prove it; if not, please explain it with counter examples; (3) try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF respectively, so that the length of the two segments is always equal when quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion
• 8. It is known that P is a point AC on the diagonal of square ABCD (not coincident with a and C), PE is perpendicular to point E, PF is perpendicular to point F 1, and CD is perpendicular to point F 1 It is known that P is a point AC (not coincident with a and C) on the diagonal of square ABCD, PE is perpendicular to point E, PF is perpendicular to point F 1. Verify BP = DP 2. If the quadrilateral PECF rotates counterclockwise around point C, is there BP = DP in the process of rotation? If so, please prove it, if not, please use counter examples Try to select two vertices of square ABCD and connect them with the two vertices of quadrilateral PECF, so that the length of the two segments is equal when the quadrilateral PECF rotates counterclockwise around point C, and prove your conclusion Thank you on your knees
• 9. The side length of square ABCD is a, PA ⊥ plane ABCD, PA = a, m, n are the midpoint of PD, Pb respectively, then the cosine value of the angle formed by the line am and cn is
• 10. As shown in the figure, a point P, PE ⊥ ad in the square ABCD is equal to E. if Pb = PC = PE = 5, then the side length of the square is______ ．
• 11. For a point in the square ABCD, rotate △ ABP clockwise 90 ° around B to the position of △ CBE, if BP = A. calculate the area of the square with PE as the side length Write every step in detail Using Pythagorean theorem
• 12. As shown in the figure, point P is a point in square ABCD. Rotate △ ABP around point B and coincide with △ BCP '. If BP = 3, find the length of PP ′ De
• 13. Point P is a point in the square ABCD, connecting PA, Pb and PC. rotate the triangle PAB 90 degrees clockwise around point B to the position of triangle p'cb Let AB be a and Pb be B (b)
• 14. P is a point in the square ABCD, and AP = 2. Rotate △ APB clockwise 60 ° around a to get △ AP ′ B ′. (1) make the rotated figure; (2) try to find the perimeter and area of △ app ′
• 15. In parallelogram ABCD, P is a point outside ad, and AP is vertical to PC, BP is vertical to PD, so find parallelogram ABCD as rectangle
• 16. As shown in the figure, in the parallelogram ABCD, ab = 4, ad = 6, ∠ ABC = 60 °, point P is a moving point on the ray ad, BP and AC intersect at point E, let AP = X 1. Find the length of AC. 2. If △ ABC and △ BCE are similar, find the value of X. 3. If △ ABC is an isosceles triangle, find the value of X
• 17. It is known that the bottom surface of the pyramid p-abcd is a right angled trapezoid, ab ‖ DC, ∠ DAB = 90 °. Please see the following It is known that the bottom of p-abcd is a right angled trapezoid, ab ‖ DC, ∠ DAB = 90 °, PA ⊥ bottom ABCD, PA = ad = DC = 1 / 2, ab = 1, e and m are the midpoint of edge PD and PC respectively 1. Verification: AE ⊥ PCD 2. Find a point n on the line AB so that the PDA of Mn ‖ plane
• 18. As shown in the figure, it is known that in trapezoidal ABCD, ad is parallel to BC, ∠ ABC = 60 °, BD bisects ∠ ABC, and BD is vertical to DC (1) Verification: trapezoid ABCD is isosceles trapezoid (2) When CD = 1, find the circumference of the isosceles trapezoid ABCD
• 19. It is known that in trapezoidal ABCD, DC | AB, ad = BC, BD bisects | ABC, | a = 60 degree Find the degree of ∠ abd If ad = 2, find the length of diagonal BD
• 20. As shown in the figure, in the trapezoidal ABCD, DC ‖ AB, ad = BC, BD bisects ∠ ABC, a = 60 ° (1) find the degree of ∠ abd; (2) if ad = 2, find the length of diagonal BD