Problem (2020CHNSouthEastMOg10p2): In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively. Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; p = on_tline b o b, on_tline c o c; r = on_circle o a, on_pline a b c; q = on_line p r, on_circle o a; i = incenter a b c; d = foot i b c; g = on_line q d, on_circle o a; m = on_tline i a i, on_line a b; n = on_tline i a i, on_line a c ? cyclic a g m n
HAGeo formalization: A B C = triangle; O = circumcenter A B C; (O) = circle_center_point O A; OB = line O B; OC = line O C; l1 = perpendicular_line B OB; l2 = perpendicular_line C OC; P = intersection l1 l2; BC = line B C; l3 = parallel_line A BC; R = intersection (O) l3; PR = line P R; Q = intersection PR (O); I = incenter A B C; D = foot I BC; DQ = line D Q; G = intersection DQ (O); AI = line A I; l4 = perpendicular_line I AI; AB = line A B; M = intersection l4 AB; AC = line A C; N = intersection l4 AC; Prove: concyclic A G M N
Difficulty score: 2.9