Problem (2023VietnamTSTp3): Let $ABC$ be an acute, non-isosceles triangle with circumcircle $(O)$. $BE, CF$ are the heights of $\triangle ABC$, and $BE, CF$ intersect at $H$. Let $M$ be the midpoint of $AH$, and $K$ be the point on $EF$ such that $HK \perp EF$. A line not going through $A$ and parallel to $BC$ intersects the minor arc $AB$ and $AC$ of $(O)$ at $P$, $Q$, respectively. Show that the tangent line of $(CQE)$ at $E$, the tangent line of $(BPF)$ at $F$, and $MK$ concur.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; e = foot b a c; f = foot c a b; h = on_line b e, on_line c f; m = midpoint a h; k = foot h e f; p = on_circle o a; q = on_circle o a, on_pline p b c; o1 = circumcenter c q e; o2 = circumcenter b p f; x = on_tline e o1 e, on_tline f o2 f ? coll x m k
HAGeo formalization: A B C = triangle; O = circumcenter A B C; AC = line A C; E = foot B AC; AB = line A B; F = foot C AB; BE = line B E; CF = line C F; H = intersection BE CF; M = midpoint A H; EF = line E F; K = foot H EF; (O) = circle_center_point O A; P = on_circle (O); BC = line B C; l1 = parallel_line P BC; Q = intersection (O) l1; O1 = circumcenter C Q E; O2 = circumcenter B P F; O1E = line O1 E; l2 = perpendicular_line E O1E; O2F = line O2 F; l3 = perpendicular_line F O2F; X = intersection l2 l3; Prove: collinear X M K
Difficulty score: 4.7