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.

Problem 5. Point M lies on diagonal BD of parallelogram ABCD. Line AM intersects side CD and line BC at points K and N, respectively. Let C_1 be the circle with center M and radius MA and C_2 be the circumcircle of triangle KCN. C_1, C_2 intersect at P and Q. Prove that MP and MQ are tangent to C_2.

Problem 6. In an acute-angled triangle ABC, AD,BE and CF are the altitudes and H the orthocentre. Lines EF and BC meet at N. The line passing through D and parallel to FE meets lines AB and AC at K and L, respectively. Prove that the circumcircle of the triangle NKL bisects the side BC.

Problem 7. Let ABC an acute triangle so that AB\not= AC and let H the orthocenter of ABC. Points D and E are in the sides AB,AC respectively so that D,H,E are collinear and AE=AD. Prove that if M is the midpoint of BC, then MH is perpendicular to the straight line that joins A with the second intersections (different of A) of the circumcircles of the triangles AED and ABC.

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