Question 165
Let $\triangle ABC$ be an acute triangle with $|AB| \gt |AC|$ and let $\omega$ be the circumcircle of $\triangle ABC$ with centre $O$. The altitude from $A$ intersects $BC$ in $D$ and intersects $\omega$ a second time in $P$. Let $H$ be the orthocentre of $\triangle ABC$ and let $K$ be the point on the line segment $BC$ such that $|BD| = |KC|$. The circumcircle of $\triangle PKH$ intersects $\omega$ a second time in $Q$ and intersects the line $BC$ a second time in $N$. Let $T$ be the point on the line $AD$ such that $TN \perp PQ$.
Prove that the line $KT$ passes through $O$.
Prove that the line $KT$ passes through $O$.