In acute triangle ABC (AB<AC) point O is center of its circumcircle Ω. Let the tangent to Ω drawn at point A intersect the line BC at point D. Let the line DO intersects the segments AB and AC at points E and F, respectively. Point G is constructed such that AEGF is a parallelogram. Let K and H be points of intersection of segment BC with segments EG and FG, respectively. Prove that the circle (GKH) touches the circle Ω.
Proposed by Dong Luu geometrycircumcircleparallelogramtangent circles