Suppose $(E,\la.,.\ra)$ is an inner product space and $P\in\Hom(E)$ is a projection onto a non-degenerate subspace $F$, i.e. $P^2=P$ and $\im P=F$. Then $P$ is the orthogonal projection onto $F$, if and only if $P^*=P$.
We already know that orthogonal projections are self-adjoint (cf. proposition). Hence it suffices to prove the converse implication: If $P^*=P$, then for all $y\in F^\perp$ and all $x\in F$: $\la Py,x\ra=\la y,Px\ra=0$, i.e. $Py\perp F$ and thus $Py\in F\cap F^\perp=\{0\}$, since $F$ is non-degenerate.