$P$ is projection matrix iff $A$ is reflection matrix? I have the following definition

An $n \times n$ matrix $A$ is a reflection matrix if and only if $A^2 = I$ and $A^T= A$. A projection matrix is $P = 1/2(A+I)$.

I was wondering if I can conclude that $P$ is a projection matrix if and only if $A$ is a reflection matrix. If it can be said can you please explain why?
Assuming that this is true can I say that $A^2 = I$ and $A^T = A$  if and only if $P^2 = P$ and $P^T=P$? 

Since you’re restricting $P$ to an orthogonal projection, consider one of the standard ways to construct the (orthogonal) reflection of a vector relative to some subspace of $\mathbb R^n$: find the orthogonal rejection of the vector from that subspace and reverse it. That is, if $W\subset\mathbb R^n$ is a subspace and $\pi_W$ is orthogonal projection onto $W$, then the reflection of a vector $v$ in $W$ is $\rho_Wv=\pi_Wv-(v-\pi_wv)=2\pi_Wv-v$, or, in matrix form, $Av=(2P-I)v$, from which your equation $P=(A+I)/2$ follows.
