This problem comes from this year's International Mathematical Olympiad , comparing with another geometry problem on the next day (problem 4) , this seems a little harder . Enjoy !
Let be the incentre of triangle and let be its circumcircle. Let the line intersect again at . Let be a point on the arc and a point on the side such that
Finally , let be the midpoint of the segment . Prove that the lines and intersects on .