Question 190
Let $O$ denote the circumcentre of $\triangle ABC$, and $M$ the midpoint of $BC$. The line perpendicular to $AC$ passing through the point $C$ intersects line $AB$ (extended) at $T$, and intersects the circumcircle of $\triangle ABC$ for the second time at $D$. Let $L$ denote a point on the line $AD$ (extended) such that $CT$ is the angle bisector of $\angle BCL$. Lines $OM$ and $TL$ intersect at $P$. Prove that the lines $PB$ and $PC$ are tangent to the circumcircle of $\triangle ABC$.