Question 50
Let $k_{1}$ be a circle centred at $M$ and $\ell$ a line tangent to $k_{1}$ at $A$. Let $k_{2}$ be a circle inside $k_{1}$ also tangent to $\ell$ at $A$. Let $P$ be a point on $\ell$ different from $A$. The second tangent to $k_{1}$ through $P$ touches $k_{1}$ at $T$. Let $B$ be the second intersection of $A T$ and $k_{2}$, and let $C$ be the second intersection of $P B$ and $k_{2}$. Show that $A T C M$ is a cyclic quadrilateral.