Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.
Source: 2020 NZMO Round 2 p4
Tags: geometry, angle bisector, tangent circles
Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.