Harmonic division (1)

Problem 1. Let \triangle ABC be a triangle. The incircle of triangle \triangle ABC touches side BC at A'. Let segment AA' meet the incircle again at P. Segments BP,CP meet the incircle at M,N, respectively. Show that lines AA',BN,CM are concurrent.

Problem 2. Given acute triangle ABC with AB>AC, let M be the midpoint of BC. P is a point in triangle AMC such that \angle MAB=\angle PAC. Let O,O_1,O_2 be the circumcenters of \triangle ABC,\triangle ABP,\triangle ACP respectively. Prove that line AO passes through the midpoint of O_1 O_2.

Problem 3. Let ABCD be a cyclic kite (i.e. BD is a perpendicular chord onto the diameter AC) and M the midpoint of AD. The perpendicular from C onto BM intersects AD at P. Prove that BP is tangent to the circle \odot (ABC).

Problem 4. In triangle ABC, let I be the incenter and let I_a be the excenter opposite A. Suppose that II_a meets BC and the circumcircle of triangle ABC at A_0 and M, respectively. Let N be the midpoint of arc MBA of the circumcircle of triangle ABC. Let lines NI and NI_a intersect the circumcircle of triangle ABC again at S and T, respectively. Prove that S, T, and A_0 are collinear. Continue reading “Harmonic division (1)”