Problem (2006CTSTp19): $H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; h = orthocenter a b c; d = on_circum a b c; e = on_circum a b c, on_pline b a d; f = on_circum a b c, on_pline c a d; s = reflect d b c; t = reflect e c a; u = reflect f a b ? cyclic s t u h
HAGeo formalization: A B C = triangle; H = orthocenter A B C; O = circumcenter A B C; (ABC) = circle A B C; D = on_circle (ABC); AD = line A D; l1 = parallel_line B AD; E = intersection (ABC) l1; l2 = parallel_line C AD; F = intersection (ABC) l2; BC = line B C; S = reflect_point_wrt_line D BC; AC = line A C; T = reflect_point_wrt_line E AC; AB = line A B; U = reflect_point_wrt_line F AB; Prove: concyclic S T U H
Difficulty score: 3.4