Problem

Source: Philippine Mathematical Olympiad 2024 P3

Tags: geometry, orthocenter



Given triangle $ABC$ with orthocenter $H$, the lines through $B$ and $C$ perpendicular to $AB$ and $AC$, respectively, intersect line $AH$ at $X$ and $Y$, respectively. The circle with diameter $XY$ intersects lines $BX$ and $CY$ a second time at $K$ and $L$, respectively. Prove that points $H, K$ and $L$ are collinear.