Problem (2020CzechAPSlovakp1): Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear.
AlphaGeometry formalization: a b c = triangle; d = parallelogram a b c d; p = on_line a c, on_line b d; m = midpoint a b; o1 = circumcenter m a d; o2 = circumcenter m b c; q = on_tline a o1 a, on_tline b o2 b ? coll q m p
HAGeo formalization: A B C D = parallelogram; AC = line A C; BD = line B D; P = intersection AC BD; M = midpoint A B; O1 = circumcenter A D M; O1A = line O1 A; l1 = perpendicular_line A O1A; O2 = circumcenter B C M; O2B = line O2 B; l2 = perpendicular_line B O2B; Q = intersection l1 l2; Prove: collinear Q M P
Difficulty score: 2.3