Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circles, internally tangent at $P$ ($\mathcal{C}_2$ lies inside of $\mathcal{C}_1$). A chord $AB$ of $\mathcal{C}_1$ is tangent to $\mathcal{C}_2$ at $C.$ Let $D$ be the second point of intersection between the line $CP$ and $\mathcal{C}_1.$ A tangent from $D$ to $\mathcal{C}_2$ intersects $\mathcal{C}_1$ for the second time at $E$ and it intersects $\mathcal{C}_2$ at $F.$ Prove that $F$ is the incenter of triangle $ABE.$