Problem (2018ARMOg11p4): On the sides $AB$ and $AC$ of the triangle $ABC$, the points $P$ and $Q$ are chosen, respectively, so that $PQ\parallel BC$. Segments $BQ$ and $CP$ intersect at point $O$. Point $A'$ is symmetric to point $A$ relative to line $BC$. The segment $A'O$ intersects the circumcircle $w$ of the triangle $APQ$ at the point $S$. Prove that circumcircle of $BSC$ is tangent to the circle $w$.
AlphaGeometry formalization: a b c = triangle; p = on_line a b; q = on_line a c, on_pline p b c; o = on_line b q, on_line c p; a1 = reflect a b c; o1 = circumcenter a p q; s = on_line a1 o, on_circle o1 a; o2 = circumcenter b c s ? coll s o1 o2
HAGeo formalization: A B C = triangle; AB = line A B; P = on_line AB; BC = line B C; l1 = parallel_line P BC; AC = line A C; Q = intersection AC l1; BQ = line B Q; CP = line C P; O = intersection BQ CP; A1 = reflect_point_wrt_line A BC; A1O = line A1 O; O1 = circumcenter A P Q; (O1) = circle_center_point O1 A; S = intersection A1O (O1); O2 = circumcenter B C S; Prove: collinear S O1 O2
Difficulty score: 3.9