Problem (2023TurkeyTSTp5): Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; i = incenter a b c; h = orthocenter a b c; o1 = on_bline o i, on_tline i i h; o2 = on_bline h i, on_tline i i o; m = on_circle o1 o, on_circle o2 h ? cyclic m a b c
HAGeo formalization: A B C = triangle; O = circumcenter A B C; I = incenter A B C; H = orthocenter A B C; l1 = perpendicular_bisector O I; IH = line I H; l2 = perpendicular_line I IH; O1 = intersection l1 l2; l3 = perpendicular_bisector H I; IO = line I O; l4 = perpendicular_line I IO; O2 = intersection l3 l4; (O1) = circle_center_point O1 O; (O2) = circle_center_point O2 H; M = intersection (O1) (O2); Prove: concyclic M A B C
Difficulty score: 4.5