Problem (2011CTSTp16): Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; o = circumcenter a b c; p = on_circle o a; m = on_line p h, on_circle o a; p1 = foot p b c; p2 = foot p c a; k = on_circle o a, on_pline m p1 p2; q = on_circle o a, on_pline p b c; j = on_line b c, on_line k q ? cong k j m j
HAGeo formalization: A B C = acute_triangle; H = orthocenter A B C; O = circumcenter A B C; (O) = circle_center_point O A; P = on_circle (O); HP = line H P; M = intersection (O) HP; BC = line B C; P1 = foot P BC; AC = line A C; P2 = foot P AC; P1P2 = line P1 P2; l1 = parallel_line M P1P2; K = intersection (O) l1; l2 = parallel_line P BC; Q = intersection (O) l2; KQ = line K Q; J = intersection BC KQ; Prove: cong J K J M
Difficulty score: 3.9