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 is just the fact that $\sqrt{2}$ is irrational. Suppose there would be such a right-triangle. Then $n=1$ would be a congruent number, i.e., the area of an right-triangle with rational sides. By Fermat, for exponent $4$, it isn't.
