Problem (2016G5): Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; h = orthocenter a b c; d = foot a o h; s = on_bline a d; x = on_circle s a, on_line a b; y = on_circle s a, on_line a c; p = foot a b c; m = midpoint b c; o1 = circumcenter x s y ? cong o1 p o1 m
HAGeo formalization: A B C = triangle; O = circumcenter A B C; H = orthocenter A B C; OH = line O H; D = foot A OH; l = perpendicular_bisector A D; S = on_line l; (S,A) = circle_center_point S A; AB = line A B; X = intersection (S,A) AB; AC = line A C; Y = intersection (S,A) AC; BC = line B C; P = foot A BC; M = midpoint B C; O1 = circumcenter X S Y; Prove: cong O1 P O1 M
Difficulty score: 4.3