Problem (2021RMMSLG4): 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.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; o = circumcenter a b c; s = foot b a c; t = foot c a b; m = midpoint s t; n = midpoint a h; f = on_pline o b c, on_line a c; g = on_pline o b c, on_line a b; o1 = circumcenter b g o; k = on_line n g, on_circle o1 b; o2 = circumcenter c f o; l = on_line n f, on_circle o2 c ? simtri b c m k l n
HAGeo formalization: A B C = acute_triangle; H = orthocenter A B C; O = circumcenter A B C; AC = line A C; S = foot B AC; AB = line A B; T = foot C AB; M = midpoint S T; N = midpoint A H; BC = line B C; l1 = parallel_line O BC; F = intersection l1 AC; l2 = parallel_line O BC; G = intersection l2 AB; GN = line G N; O1 = circumcenter B G O; (BGO) = circle_center_point O1 B; K = intersection GN (BGO); FN = line F N; O2 = circumcenter C F O; (CFO) = circle_center_point O2 C; L = intersection FN (CFO); Prove: similar B C M K L N
Difficulty score: 4.7