This is a brilliant proof which requires only paper folding. The idea is to think of the smallest possible integer sides isosceles triangle and one is able to come up with a still smaller integer side triangle and thus contradicting our assumption that such a triangle is possible.
All one has to do is project the right angled vertex on the hypotenuse, in the process marking off the isosceles side on the longest side and this carves out a smaller similar triangle. The proof is in the same spirit as the previous proof where we we had to extend the isosceles side to hypotenuse and then drawing a projection of this extended line back on hypotenuse.
Sumant's 1 page of Math 