Problem (2020AQGOp1): Let $ABC$ be a triangle with altitude $AD$. Let $V$ be a variable point on $(ABC)$ and $\overline{VD}$ intersect $(ABC)$ at $U$. Let $\overline{UM}$ intersect $(ABC)$ at $X$ where $M$ is the midpoint of $BC$. Let $\overline{AX}\cap\overline{BC}=Y$. Let $\odot(YVM)$ intersect $(ABC)$ at a point $Z$. Let $Z'$ be the reﬂection of $Z$ over $\overline{OH}$. Finally let $(Z'OH)$ intersect $(ABC)$ at $T\ne Z'$. Then prove that $TZ$ passes through $M$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; h = orthocenter a b c; d = foot a b c; v = on_circle o a; u = on_line v d, on_circle o a; m = midpoint b c; x = on_line u m, on_circle o a; y = on_line a x, on_line b c; o1 = circumcenter y v m; z = on_circle o1 y, on_circle o a; z1 = reflect z o h; o2 = circumcenter z1 o h; t = on_circle o2 z1, on_circle o a ? coll t z m
HAGeo formalization: A B C = triangle; O = circumcenter A B C; H = orthocenter A B C; BC = line B C; D = foot A BC; (O) = circle_center_point O A; V = on_circle (O); DV = line D V; U = intersection DV (O); M = midpoint B C; MU = line M U; X = intersection MU (O); AX = line A X; Y = intersection AX BC; O1 = circumcenter M V Y; (O1) = circle_center_point O1 Y; Z = intersection (O1) (O); HO = line H O; Z1 = reflect_point_wrt_line Z HO; O2 = circumcenter H O Z1; (O2) = circle_center_point O2 Z1; T = intersection (O2) (O); Prove: collinear T Z M
Difficulty score: 3.7