Continuing about Pythagorean triples, the tutor considers the isosceles case.

In my previous post I discussed all-integer solutions to the Pythagoras equation

a^2 + b^2 =c^2

Such solutions are often called Pythagorean triples.

Presently we consider the possibility of Pythagorean triples that have two equal numbers – that is, integer solutions to

x^2 +x^2 = y^2

The equation simplifies to

2x^2 = y^2

Dividing both sides by x^2 gives

2=y^2/x^2

Square rooting both sides, we arrive at

2^(1/2)=y/x

Since 2^(1/2) is not a rational number,

2^(1/2)=y/x

has no integer solutions for both x and y.

Therefore, there are no (all-integer) Pythagorean triples of the form x,x,y.

Source:

Dudley, Underwood. Elementary Number Theory, 2nd Ed. New York: W H Freeman
   and Company, 1978.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.