Problem (2008ARMOg9p3): In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; x y z m = centroid a b c; p = on_tline b b m, on_tline c c m; q = on_tline c c m, on_tline a a m; r = on_tline a a m, on_tline b b m; x1 y1 z1 m1 = centroid p q r ? coll m1 m h
HAGeo formalization: A B C = triangle; H = orthocenter A B C; M X Y Z = centroid A B C; AM = line A M; l1 = perpendicular_line A AM; BM = line B M; l2 = perpendicular_line B BM; CM = line C M; l3 = perpendicular_line C CM; P = intersection l2 l3; Q = intersection l1 l3; R = intersection l1 l2; M1 X1 Y1 Z1 = centroid P Q R; Prove: collinear M H M1
Difficulty score: 3.8