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circle touching circumcircle at A & touching externally its incircle

Source: Sharygin Geometry Olympiad 2016 First Round P14 grades 9-11 (a for 9-10, b for 10-11)

July 25, 2018
geometrysymmetrysymmediantangent circlescircumcircleincircleconcurrency

Problem Statement

Let a triangle ABCABC be given. Consider the circle touching its circumcircle at AA and touching externally its incircle at some point A1A_1. Points B1B_1 and C1C_1 are defined similarly. a) Prove that lines AA1,BB1AA_1, BB_1 and CC1CC1 concur. b) Let A2A_2 be the touching point of the incircle with BCBC. Prove that lines AA1AA_1 and AA2AA_2 are symmetric about the bisector of angle ∠A\angle A.
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