Problem (2018CMOp4): $ABCD$ is a cyclic quadrilateral whose diagonals intersect at $P$. The circumcircle of $\triangle APD$ meets segment $AB$ at points $A$ and $E$. The circumcircle of $\triangle BPC$ meets segment $AB$ at points $B$ and $F$. Let $I$ and $J$ be the incenters of $\triangle ADE$ and $\triangle BCF$, respectively. Segments $IJ$ and $AC$ meet at $K$. Prove that the points $A,I,K,E$ are cyclic.
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; p = on_line a c, on_line b d; o1 = circumcenter a p d; e = on_line a b, on_circle o1 a; o2 = circumcenter b p c; f = on_line a b, on_circle o2 b; i = incenter a d e; j = incenter b c f; k = on_line i j, on_line a c ? cyclic a i k e
HAGeo formalization: A B C = triangle; (ABC) = circle A B C; D = on_circle (ABC); AC = line A C; BD = line B D; P = intersection AC BD; O1 = circumcenter A P D; (ADP) = circle_center_point O1 A; AB = line A B; E = intersection (ADP) AB; O2 = circumcenter B P C; (BCP) = circle_center_point O2 B; F = intersection (BCP) AB; I = incenter A D E; J = incenter B C F; IJ = line I J; K = intersection IJ AC; Prove: concyclic A I K E
Difficulty score: 3.2