Problem (2012EGMOp7): Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$. Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; h = orthocenter a b c; k = on_circle o a; l = reflect k a b; m = reflect k b c; o1 = circumcenter b l m; e = on_circle o b, on_circle o1 b; p = on_line k h, on_line e m ? coll p b c
HAGeo formalization: A B C = acute_triangle; O = circumcenter A B C; (O) = circle_center_point O A; H = orthocenter A B C; K = on_circle (O); AB = line A B; L = reflect_point_wrt_line K AB; BC = line B C; M = reflect_point_wrt_line K BC; O1 = circumcenter B L M; (O1) = circle_center_point O1 B; E = intersection (O) (O1); HK = line H K; EM = line E M; P = intersection HK EM; Prove: collinear P B C
Difficulty score: 2.9