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Geometry Mathley 4.3 concurrent lines, tangent and concurrent circles

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June 7, 2020
geometryconcurrenttangent circlescirclesincirclecircumcircle

Problem Statement

Let ABCABC be a triangle not being isosceles at AA. Let (O)(O) and (I)(I) denote the circumcircle and incircle of the triangle. (I)(I) touches ACAC and ABAB at E,FE, F respectively. Points MM and NN are on the circle (I)(I) such that EM∥FN∥BCEM \parallel FN \parallel BC. Let P,QP,Q be the intersections of BM,CNBM,CN and (I)(I). Prove that i) BC,EP,FQBC,EP, FQ are concurrent, and denote by KK the point of concurrency. ii) the circumcircles of triangle BPK,CQKBPK, CQK are all tangent to (I)(I) and all pass through a common point on the circle (O)(O).
Nguyễn Minh Hà
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