It is known that, as shown in the figure, P is a point in the square ABCD, and there is a point e outside the square ABCD, satisfying the angle Abe = angle CBP, be = BP 2.PB⊥BE.

It is known that, as shown in the figure, P is a point in the square ABCD, and there is a point e outside the square ABCD, satisfying the angle Abe = angle CBP, be = BP 2.PB⊥BE.

prove:
A quadrilateral ABCD is a square
∴BA=BC
∵∠ABE=∠CBP,BE=BP
∴△ABE≌△ABP（SAS）

P is a point inside the square ABCD, and there is a point e outside the square ABCD, satisfying the angle Abe = angle CBP, be = BP, if PA = 2PB, ∠ APB = 135 °, calculate AP: AE

Solution: connecting PE in △ Abe and △ CBP, be = BP, ∠ Abe = ∠ CBP, ab = BC ≌ Abe ≌ CBP (SAS) and