Problem (2021AsiaPacificMOp3): Let $ABCD$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals of $AC$ and $BD$. Let $L$ be the center of the circle tangent to sides $AB$, $BC$, and $CD$, and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $BCE$ opposite $E$ lies on the line $LM$.
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; o = circumcenter a b c; e = on_line a c, on_line b d; x = on_line a b, on_line c d; l = incenter x b c; m = on_bline b c, on_circle o a; i1 = excenter e b c ? coll i1 l m
HAGeo formalization: A B C = triangle; O = circumcenter A B C; (ABC) = circle_center_point O A; D = on_circle (ABC); AC = line A C; BD = line B D; E = intersection AC BD; AB = line A B; CD = line C D; X = intersection AB CD; L = incenter X B C; l1 = perpendicular_bisector B C; M = intersection l1 (ABC); Ie = excenter E B C; Prove: collinear Ie L M
Difficulty score: 3.0