Pythagorean Triple where $a=b$? I cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to use $ a = 10 $, $ a = 10 $, $ c = 14 $ because $ 10 \sqrt{2}  $ is close enough to $ 14 $. In my program I am worried this could have serious consequences because it's not accurate. 
Does there exist a case where $ 2 a^2 = c^2 $ where a and c are whole numbers? 
If it does not exist, why? Does it revolve around the fact that $ \sqrt{2} $ is irrational?

Yes, it does revolve around the fact that $\sqrt{2}$ is irrational.
For if there were integers $a$ and $c$ such that $a^2 + a^2 = c^2$, then $2a^2 = c^2$, or $2 = \left(\frac{c}{a}\right)^2$.  Therefore $\sqrt{2}$ is rational $\Rightarrow\Leftarrow$.
