Pythagorean Triangle cannot be Isosceles

Theorem
Let $P$ be a Pythagorean triangle.

Then $P$ is not isosceles.

Theorem
Let $P$ be a Pythagorean triangle.

$P$ is an isosceles.

Let the legs of $P$ be of length $a$.

Let the hypotenuse of $P$ be of length $h$.

We have from Pythagoras's Theorem that:
 * $2 a^2 = h^2$

and so:
 * $\dfrac h a = \sqrt 2$

By definition, $h$ and $a$ are integers.

Hence, by definition, $\sqrt 2$ is a rational number.

But that contradicts the result Square Root of 2 is Irrational.

By Proof by Contradiction, it follows that $P$ cannot be isosceles.