Problem (ShuZhiMi2023SpFp8): For $\triangle ABC$ with circumcenter $O$, incenter $I$, orthocenter $H$, and $A$-excenter $J$, let $A_1$ be $A$'s reflection over $BC$, $A_2=AO\cap(BOC)\setminus A$, $R=OI\cap JA_1$, $Q=IH\cap JA_2$, and $P$ be the second intersection of $(AOH)$'s tangent at $A$ with $(ABC)$. Prove $Q,P,R$ are collinear.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; i = incenter a b c; h = orthocenter a b c; j = excenter a b c; a1 = reflect a b c; o1 = circumcenter b o c; a2 = on_line a o, on_circle o1 b; r = on_line o i, on_line j a1; q = on_line i h, on_line j a2; o2 = circumcenter a o h; p = on_tline a o2 a, on_circle o a ? coll q p r
HAGeo formalization: A B C = triangle; O = circumcenter A B C; I = incenter A B C; H = orthocenter A B C; J = excenter A B C; BC = line B C; A1 = reflect_point_wrt_line A BC; O1 = circumcenter B O C; AO = line A O; (O1) = circle_center_point O1 B; A2 = intersection AO (O1); OI = line O I; JA1 = line J A1; R = intersection OI JA1; IH = line I H; JA2 = line J A2; Q = intersection IH JA2; O2 = circumcenter A O H; O2A = line O2 A; l1 = perpendicular_line A O2A; (O) = circle_center_point O A; P = intersection l1 (O); Prove: collinear Q P R
Difficulty score: 6.5