Problem 1. Let be a triangle. The incircle of triangle touches side at Let segment meet the incircle again at Segments meet the incircle at respectively. Show that lines are concurrent.
Problem 2. Given acute triangle with , let be the midpoint of . is a point in triangle such that . Let be the circumcenters of respectively. Prove that line passes through the midpoint of .
Problem 3. Let be a cyclic kite (i.e. is a perpendicular chord onto the diameter ) and the midpoint of . The perpendicular from onto intersects at . Prove that is tangent to the circle .
Problem 4. In triangle , let be the incenter and let be the excenter opposite . Suppose that meets and the circumcircle of triangle at and , respectively. Let be the midpoint of arc of the circumcircle of triangle . Let lines and intersect the circumcircle of triangle again at and , respectively. Prove that , , and are collinear. Continue reading “Harmonic division (1)”