SHARYGIN 2006 Problem 19

Through the midpoints of the sides of the triangle $T$ , straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$ . Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$ .