Question 166
Consider three points $A$, $B$, $C$ on a circle $\Gamma$, with $\angle BAC \gt 90^\circ$. Let $d$ denote the line tangent to $\Gamma$ at $A$. Points $M$ and $N$ are chosen on $d$ such that $\angle MBA = \angle ABC$ and $\angle NCA = \angle ACB$.
Prove that $A$ is the midpoint of segment $MN$.
Prove that $A$ is the midpoint of segment $MN$.