Question 54
The circle $\mathcal{C}_1$ is inside the circle $\mathcal{C}_2$, and the circles touch each other at $A$. A line through $A$ intersects $\mathcal{C}_1$ also at $B$ and $\mathcal{C}_2$ also at $C$. The tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $D$ and $E$. The tangents of $\mathcal{C}_1$ passing through $C$ touch $\mathcal{C}_1$ at $F$ and $G$. Prove that $D$, $E$, $F$, and $G$ are concyclic.