Problem (2015BalkanMOp2): Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; o = circumcenter a b c; d = on_line a i, on_circle o a; e = on_line b i, on_circle o b; f = on_line c i, on_circle o c; k = on_line e f, on_pline i b c; l = on_line d f, on_pline i a c; m = on_line d e, on_pline i a b ? coll k l m
HAGeo formalization: A B C = triangle; I = incenter A B C; O = circumcenter A B C; (O) = circle_center_point O A; AI = line A I; D = intersection AI (O); BI = line B I; E = intersection BI (O); CI = line C I; F = intersection CI (O); BC = line B C; l1 = parallel_line I BC; EF = line E F; K = intersection l1 EF; AC = line A C; l2 = parallel_line I AC; DF = line D F; L = intersection l2 DF; AB = line A B; l3 = parallel_line I AB; DE = line D E; M = intersection l3 DE; Prove: collinear K L M
Difficulty score: 1.9