Let ABC be a triangle not being isosceles at A. Let (O) and (I) denote the circumcircle and incircle of the triangle. (I) touches AC and AB at E,F respectively. Points M and N are on the circle (I) such that EM∥FN∥BC. Let P,Q be the intersections of BM,CN and (I). Prove that
i) BC,EP,FQ are concurrent, and denote by K the point of concurrency.
ii) the circumcircles of triangle BPK,CQK are all tangent to (I) and all pass through a common point on the circle (O).Nguyễn Minh Hà geometryconcurrenttangent circlescirclesincirclecircumcircle