We have a Pythagorean triple of the form $\displaystyle a^2+b^2=c^2$ with $\displaystyle a,b,c$ being positive integers. I have to show that $\displaystyle c^2 + \frac{2}{3}ab$ is a composite integer.

I get winded up in some extra long derivations that force me to make even more substitutions, but I don't really get anything that would lead me to reasonable conclusions.