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Incircle Angle chase geo

Source: Sharygin Correspondence Round 2024 P4

March 6, 2024
geometry

Problem Statement

The incircle ω\omega of triangle ABCABC touches BC,CA,ABBC, CA, AB at points A1,B1A_1, B_1 and C1C_1 respectively, PP is an arbitrary point on ω\omega. The line APAP meets the circumcircle of triangle AB1C1AB_1C_1 for the second time at point A2A_2. Points B2B_2 and C2C_2 are defined similarly. Prove that the circumcircle of triangle A2B2C2A_2B_2C_2 touches ω\omega.
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