Question 101
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent at the point $P$ (the circle $\mathcal{C}_2$ lies inside the circle $\mathcal{C}_1$).
A chord $AB$ of the circle $\mathcal{C}_1$ is tangent to the circle $\mathcal{C}_2$ at the point $C$. Let $D$ be the second intersection point of the line $CP$ with the circle $\mathcal{C}_1$. A tangent drawn from $D$ to $\mathcal{C}_2$ meets the circle $\mathcal{C}_1$ again at $E$ and the circle $\mathcal{C}_2$ at $F$.
Show that the point $F$ is the incenter of the triangle $ABE$.
A chord $AB$ of the circle $\mathcal{C}_1$ is tangent to the circle $\mathcal{C}_2$ at the point $C$. Let $D$ be the second intersection point of the line $CP$ with the circle $\mathcal{C}_1$. A tangent drawn from $D$ to $\mathcal{C}_2$ meets the circle $\mathcal{C}_1$ again at $E$ and the circle $\mathcal{C}_2$ at $F$.
Show that the point $F$ is the incenter of the triangle $ABE$.