Pythagorean Triangle cannot be Isosceles

Theorem

Let $P$ be a Pythagorean triangle.

Then $P$ is not isosceles.

Theorem

Let $P$ be a Pythagorean triangle.

Aiming for a contradiction, suppose $P$ 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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$