The incircle of an acute-angled triangle ABC touches AB,BC,CA at points C1,A1,B1 respectively. Points A2,B2 are the midpoints of the segments B1C1,A1C1 respectively. Let P be a common point of the incircle and the line CO, where O is the circumcenter of triangle ABC. Let also A′ and B′ be the second common points of PA2 and PB2 with the incircle. Prove that a common point of AA′ and BB′ lies on the altitude of the triangle dropped from the vertex C. geometrycircumcirclegeometry unsolved