Question

Prove that if all sides of a parallelogram are equal, them each diagonal is the perpendicular bisector of the other.

Answer 1

The diagonal BD divides the parallelogram into two isosceles triangles. [The angles opposite to the equal sides in an isosceles triangles are equal.] 

So the diagonal DB bisect ∠D and ∠B.

Similarly the diagonal AC bisect A and ∠C.

4x + 4y = 360° ⇒ x + y = 90°

The four triangles formed by intersecting the diagonals are equal triangles. Each one 90° angle.

So each diagonal is the perpendicular bisector of the other.

In ∆AMD ∠AMD = 180 – (x – y) = 180 – 90 = 90° 

⇒ BD ⊥ AC