Problem (2025IranTSTp12): In a scalene triangle $ABC$, points $Y$ and $X$ lie on $AC$ and $BC$ respectively such that $BC \perp XY$. Points $Z$ and $T$ are the reflections of $X$ and $Y$ with respect to the midpoints of sides $BC$ and $AC$, respectively. Point $P$ lies on segment $ZT$ such that the circumcenter of triangle $XZP$ coincides with the circumcenter of triangle $ABC$. Prove that the nine-point circle of triangle $ABC$ passes through the midpoint of segment $XP$.
AlphaGeometry formalization: a b c = triangle; y = on_line a c; x = on_line b c, on_tline y b c; m1 m2 m3 n = ninepoints a b c; z = mirror x m1; t = mirror y m2; o = circumcenter a b c; p = on_line z t, on_circle o x; m4 = midpoint x p ? cyclic m4 m1 m2 m3
HAGeo formalization: A B C = triangle; AC = line A C; Y = on_line AC; BC = line B C; l1 = perpendicular_line Y BC; X = intersection BC l1; N M1 M2 M3 = ninepointcenter A B C; Z = reflect_point_wrt_point X M1; T = reflect_point_wrt_point Y M2; O = circumcenter A B C; ZT = line Z T; (O,X) = circle_center_point O X; P = intersection ZT (O,X); M4 = midpoint X P; Prove: concyclic M4 M1 M2 M3
Difficulty score: 4.3