To prove an "iff" statement, you need to prove it both ways.

So, you need to show:

1. If the right angle triangle is isosceles then the area =

and

2. If the area then the right angle triangle is isosceles.

1. The area of the triangle is given by

.

Since the triangle is isosceles, we have

.

So .

By Pythagoras' Theorem, we have

.

So .

Try to prove Part 2...