Problem

Source: RMM Extralist 2021 G4

Tags: geometry, config geo, RMM Shortlist



Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively, and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively. Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$ meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that the triangles $BCM$ and $KLN$ are similar.