MathDB

P14

Part of 2016 Sharygin Geometry Olympiad

Problems(1)

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)

7/25/2018
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.
geometrysymmetrysymmediantangent circlescircumcircleincircleconcurrency