Question 176
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 $\angle NCA = \angle ACB$. The circumcircles of triangles $ABM$ and $ACN$ intersect at $A$ and $T$. Prove the following:
a. $TA$ is the angle bisector of $\angle BTC$.
b. The circumcircles of triangles $TMN$ and $TBC$ are tangent at $T$.
a. $TA$ is the angle bisector of $\angle BTC$.
b. The circumcircles of triangles $TMN$ and $TBC$ are tangent at $T$.