MEMO 2017 ekipno problem 5
Let be an acute-angled triangle with and circumcircle . Let be the midpoint of the shorter arc of , and let be the intersection of the rays and . Let be the intersection of the internal bisector of the angle and the circumcircle of the triangle . Let us assume that is inside the triangle and there is an intersection of the line and the circle such that is the midpoint of the segment .
Show that is the midpoint of the segment , where and are the excentres of opposite to and , respectively.