Problem (2015ARMOg11p7): A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; d = on_line a b, on_tline c o c; i = incenter a b c; q = on_line a i, angle_bisector c d b; p = on_line b i, angle_bisector c d b; m = midpoint q p; n = on_circle o a, on_tline c c i ? coll m i n
HAGeo formalization: A B C = triangle; O = circumcenter A B C; OC = line O C; l1 = perpendicular_line C OC; AB = line A B; D = intersection l1 AB; I = incenter A B C; l2 = angle_bisector C D B; AI = line A I; Q = intersection l2 AI; BI = line B I; P = intersection l2 BI; M = midpoint Q P; N = midarc A C B; Prove: collinear M I N
Difficulty score: 2.6