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

The solution is shown in Fig,

RELATED INFORMATIONS

1. 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
2. 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 ′
3. 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)
4. 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
5. 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
6. 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
7. 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,
8. 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?
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. In rectangular trapezoid ABCD, AD / / BC, ∠ ABC = 90 °, de ⊥ AC at point G, intersects the extension point e of AB, and AE = AC, if ad = DC = 2, find the length of DF How to prove that the DAC is 30 degree
16. The bottom of p-abcd is a rectangle, the side pad is an equilateral triangle, and the side pad ⊥ bottom ABCD, When the value of AD / AB is equal to, it can make Pb ⊥ AC? And the proof is given
17. In a pyramid p-abcd, planar pad ⊥ planar ABCD, ab ∥ CD, △ pad are equilateral triangles. It is known that BD = 2ad = 8, ab = 2dc = 4 √ 5 Let m be a point on PC, and prove that planar MBD ⊥ planar pad
18. As shown in the figure, EF is the two points AE parallel CF on the diagonal BD of the quadrilateral ABCD, AE = CF, be = DF to prove the congruent triangle CBF of the triangle ade
19. The vertex a of the square ABCD is in the shape ∠ EAF = 45 °, e, f is in the shape BC.CD Let EF be ah ⊥ EF in H. prove ah = ab
20. As shown in the figure, in square ABCD, points E and F are on edge BC and CD respectively. If AE = 4, EF = 3 and AF = 5, then the area of square ABCD is equal to () A. 22516B. 25615C. 25617D. 28916