Question 29
Let $\Gamma$ be the circumcircle of the acute triangle $ABC$ and $P$ a point on the arc $BC$ not containing $A$. Let $K$, $L$ and $S$ be the feet of the perpendiculars from $P$ to the lines $AB$, $AC$ and $BC$ respectively. Let $M \neq P$ and $N \neq P$ be the intersections of $PK$ and $PL$ with $\Gamma$ respectively, and $T$ the intersection of the lines $KL$ and $MN$. Prove that $OS = OT$, where $O$ is the centre of $\Gamma$.