Problem (2023IZhautykovOp2): The tangent at $C$ to $\Omega$, the circumcircle of scalene triangle $ABC$ intersects $AB$ at $D$. Through point $D$, a line is drawn that intersects segments $AC$ and $BC$ at $K$ and $L$ respectively. On the segment $AB$ points $M$ and $N$ are marked such that $AC \parallel NL$ and $BC \parallel KM$. Lines $NL$ and $KM$ intersect at point $P$ lying inside the triangle $ABC$. Let $\omega$ be the circumcircle of $MNP$. Suppose $CP$ intersects $\omega$ again at $Q$. Show that $DQ$ is tangent to $\omega$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; d = on_tline c o c, on_line a b; k = on_line a c; l = on_line d k, on_line b c; n = on_line a b, on_pline l a c; m = on_line a b, on_pline k b c; p = on_line n l, on_line k m; o1 = circumcenter m n p; q = on_line c p, on_circle o1 m ? perp d q o1 q
HAGeo formalization: A B C = triangle; O = circumcenter A B C; OC = line O C; l1 = perpendicular_line C OC; AB = line A B; D = intersection l1 AB; AC = line A C; K = on_line AC; BC = line B C; DK = line D K; L = intersection DK BC; l2 = parallel_line K BC; M = intersection AB l2; l3 = parallel_line L AC; N = intersection AB l3; LN = line L N; KM = line K M; P = intersection LN KM; O1 = circumcenter M N P; (O1) = circle_center_point O1 M; CP = line C P; Q = intersection CP (O1); DQ = line D Q; O1Q = line O1 Q; Prove: perpendicular DQ O1Q
Difficulty score: 3.0