PQRS is a square. The diagonals intersect at E. In the middle of line PS,there are diagonals A and B which also intersect at E,forming 3 triangles in triangle PES,and PA=PS?prove that AED is an isos?

To prove that triangle AED is an isosceles triangle, we need to show that AE = ED.

Since A and B are the midpoints of PS, we know that PA = AS and PB = BS. Therefore, triangle PAS is congruent to triangle PBS by the Side-Angle-Side postulate.

Since triangle PAS is congruent to triangle PBS, we know that angle PAB is congruent to angle PBS. Similarly, angle SAB is congruent to angle ASB.

Since angle PAB is congruent to angle PBS, we know that angle PAB is congruent to angle SAB. Similarly, angle ASB is congruent to angle BS.

Therefore, triangle AEP is congruent to triangle EBP by the Angle-Angle-Side postulate.

Since triangle AEP is congruent to triangle EBP, we know that AE = EB.

Since AE = EB and AE = ED, we can conclude that triangle AED is an isosceles triangle.

Since PS is a diagonal of a square, PS bisects ∠QPR. Therefore, ∠QPS = ∠SPR = 45°.

Since A is the midpoint of PS, PA = AS = PS/2.

In triangle PAE, ∠PAE = 90° - ∠QPS = 90° - 45° = 45°.

In triangle SAE, ∠ASE = 90° - ∠SPR = 90° - 45° = 45°.

Therefore, ∠PAE = ∠SAE = 45°.

Since the sum of the angles in a triangle is 180°, ∠AED = 180° - ∠PAE - ∠SAE = 180° - 45° - 45° = 90°.

Therefore, AED is an isosceles triangle.