Problem (2020CSMGOp5): Let $\triangle ABC$ be a scalene triangle with the incenter $I$, and the circumcenter $O$. Let $BI$ meet $AC$ at $D$, and let $CI$ meet $AB$ at $E$. Let the line $AI$ intersect the circumcircle of $\triangle ABC$ again at $M\ne A$, and let the perpendiculars bisectors of $BD$ and $CE$ meet at $V$. Suppose that point $X$ is placed such that $XO=XM$ and $XO\perp MV$. Prove that $XA=XI$.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; o = circumcenter a b c; d = on_line b i, on_line a c; e = on_line c i, on_line a b; m = on_line a i, on_circle o a; v = on_bline b d, on_bline c e; x = on_bline o m, on_tline o m v ? cong x a x i
HAGeo formalization: A B C = triangle; I = incenter A B C; O = circumcenter A B C; BI = line B I; AC = line A C; D = intersection BI AC; CI = line C I; AB = line A B; E = intersection CI AB; AI = line A I; (ABC) = circle_center_point O A; M = intersection AI (ABC); BD = line B D; l1 = perpendicular_bisector B D; CE = line C E; l2 = perpendicular_bisector C E; V = intersection l1 l2; l3 = perpendicular_bisector O M; MV = line M V; l4 = perpendicular_line O MV; X = intersection l3 l4; Prove: cong A X I X
Difficulty score: 4.3