Question 111
Altitudes of an acute scalene triangle $ABC$ meet at point $H$. Points $M$ and $N$ are the midpoints of the segments $AB$ and $CH$, respectively. $R$ is the foot of the perpendicular from $H$ to line $CM$. $T$ is the second intersection of the line $MN$ with the circumcircle of triangle $CNR$. Let $P$ be the circumcentre of the triangle formed by the lines $CM$, $AH$ and $BH$. Prove that the lines $HP$ and $CT$ are perpendicular.