Let $\triangle ABC$ be an acute-angled triangle. Let $H$ be the foot of the perpendicular from A to BC. Let $K$ be the foot of the the perpendicular of $H$ to $AB$, let $L$ be the foot of the perpendicular from $H$ to $AC$. Let $AH$ intersect the circumcircle of $ \triangle ABC$ in $T$, and let the line through $K$ and $L$ intersect the circumcircle of $\triangle ABC$ in $P$ and $Q$. Prove that $H$ is the incenter of $\triangle PQT$.
In any triangle the orthocenter and the circumcenter are isogonal conjugates. Since $AKHL$ is a cyclic quadrilateral, this implies that the perpendicular to $PQ$ through $A$ passes through the circumcenter of $ABC$, $O$. This implies $\widehat{POA}=\widehat{AOQ}$, so $\widehat{PTA}=\widehat{ATQ}$, and $AP=AQ$. Let $A'$ be the symmetric of $H$ with respect to $A$. If we manage to prove $AP=AH$, then the perpendicular $l_P$ to $HP$ through $P$ and the perpendicular $l_Q$ to $HQ$ through $Q$ meet on $A'$, so $PTQ$ is the orthic triangle of the triangle $\Delta$ delimited by $l_P,l_Q$ and the perpendicular $l_T$ to $HT$ through $T$. This gives that $H$ is the orthocenter of $\Delta$, hence the incenter of $PTQ$.
The best way I found so far to prove $AP=AH$ is to use some trigonometry, but I think there are more clever ways.
Edit: Thanks to wowlolbrommer, here it is a clever way: $AP=AQ$ implies $\widehat{AQL}=\widehat{ACQ}$, hence $AQ^2=AL\cdot AC$. By Euclid's first theorem, $AL\cdot AC = AH^2$, hence $AP=AQ=AH$ and we're done.
