As shown in Figure 1, slide the right angle vertex P of a right angle triangle plate on the diagonal BD of square ABCD, and make one right angle edge pass through point a all the time, and the other right angle edge intersects with BC at point E (1) Verification: PA = PE; (2) if the square in (1) is changed into a rectangle, and other conditions remain unchanged (as shown in Figure 2), and ad = 10, DC = 8, calculate AP: PE; (3) under the condition of (2), when p slides to the extension line of BD (as shown in Figure 3), please write the ratio of AP: PE directly

As shown in Figure 1, slide the right angle vertex P of a right angle triangle plate on the diagonal BD of square ABCD, and make one right angle edge pass through point a all the time, and the other right angle edge intersects with BC at point E (1) Verification: PA = PE; (2) if the square in (1) is changed into a rectangle, and other conditions remain unchanged (as shown in Figure 2), and ad = 10, DC = 8, calculate AP: PE; (3) under the condition of (2), when p slides to the extension line of BD (as shown in Figure 3), please write the ratio of AP: PE directly

(1) It is proved that: PM ⊥ AB is in M, PN ⊥ BC is in N through P, ∫ quadrilateral ABCD is a square, ∫ abd = 45 °, ∧ MPB = 45 ° = ∨ abd, ∨ PM = BM, similarly, BP = BN, ∨ quadrilateral ABCD is a square, ∨ ABC = 90 ° = ∨ BMP = ∨ BNP, ∨ quadrilateral bmpn is a square, ∨ PM = PN, ∨ MPN = 90 °, ∨ ape = 90 °, ∨ all subtract ∨ MPE to get: ∨ APM = ∨ NPE, ∨ PM ⊥ AB, PN ⊥ BC, ∨ In △ APM and △ EPN, in △ APM and △ EPN, the \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\adc = adcd = 108 = 54, ∵ ∠ amp = ∠ ENP = 90 °, ∠ M PA=∠EPN，∴△APM∽△EPN，∴APPE=PMPN=54，AP：PE=5：4；（3）AP：PE=5：4．