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A_2, B_2,C_2 cannot all lie strictly inside the circumcircle of triangle ABC

Source: IMO 2019 SL G4

September 22, 2020
geometryIMO ShortlistIMO Shortlist 2019Triangle

Problem Statement

Let PP be a point inside triangle ABCABC. Let APAP meet BCBC at A1A_1, let BPBP meet CACA at B1B_1, and let CPCP meet ABAB at C1C_1. Let A2A_2 be the point such that A1A_1 is the midpoint of PA2PA_2, let B2B_2 be the point such that B1B_1 is the midpoint of PB2PB_2, and let C2C_2 be the point such that C1C_1 is the midpoint of PC2PC_2. Prove that points A2,B2A_2, B_2, and C2C_2 cannot all lie strictly inside the circumcircle of triangle ABCABC.
(Australia)
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