Question 11
Let $k$ be a circle centred at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangents on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersects the shorter arc $\overarc{AB}$ at the point $C$ and the longer arc $\overarc{AB}$ at the point $D$, so that $|BC| = |BM|$.
Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.
Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.