Problem (2011G5): Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; o = circumcenter a b c; d = on_line a i, on_circle o a; e = on_line b i, on_circle o b; f = on_line d e, on_line a c; g = on_line d e, on_line b c; p = on_pline f a d, on_pline g b e; k = on_tline a o a, on_tline b o b; x = on_line a e, on_line b d ? coll x k p
HAGeo formalization: A B C = triangle; I = incenter A B C; O = circumcenter A B C; (O) = circle_center_point O A; AI = line A I; D = intersection (O) AI; BI = line B I; E = intersection (O) BI; DE = line D E; AC = line A C; F = intersection DE AC; BC = line B C; G = intersection DE BC; AD = line A D; l1 = parallel_line F AD; BE = line B E; l2 = parallel_line G BE; P = intersection l1 l2; OA = line O A; OB = line O B; l3 = perpendicular_line A OA; l4 = perpendicular_line B OB; K = intersection l3 l4; AE = line A E; BD = line B D; X = intersection AE BD; Prove: collinear X K P
Difficulty score: 3.1