Prove: If $\displaystyle {A^T}A = A$, then $\displaystyle A$ is symmetric & $\displaystyle A = {A^2}$

I figured out how to prove A is symmetric:

$\displaystyle {({A^T}A)^T} = {A^T}{({A^T})^T} = {A^T}A$

But how do I show that $\displaystyle A = {A^2}$ ?