Irrational Numbers

Even + Even = Even

Odd + Odd = Even

If the side is even, then its square has an area that is even.

If the side is odd, then its square has an area that is odd.

One thing that was discovered, was that if we add the area of the legs squared. It is always even.

If we had a solution then the hypotenuse can't be odd, because Odd * Odd = Odd. That would contradict the fact that the area must always be even.

Since our solution needs to have one side that is odd and the hypotenuse is even. It must be the legs.

We also need to remember that half of the area of the hypotenuse squared is the area of the leg squared.

Dividing it in half. We know that one side is at least a whole number, it may be even or odd. But the other side is definitely even.

However, this means that the area of the leg squared is even. Which means that the leg is even. However we just proved it had to be odd.

Thus we can say that this triangle does not exist, because we found a contradiction.