Prove that no isosceles right triangle exists whose sides are integers.

Answer 1

let the shorter sides be x units each ,
let the hypotenuse be h
(clearly we can't have the equal sides being the hyppotenuses, or else we would have 2 right angles, leaving nothing left for the third angle)

h^2 = x^2 + x^2
h^2 = 2x^2
h = √2x
so whatever integer x is, multiplying an integer by √ makes it irrational, thus no longer an integer.