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11 labelled points and a harmonic length condition

Source: 2019 Philippine IMO TST1 Problem 2

May 4, 2022
geometryharmonic bundlescircumcirclegeometry unsolved

Problem Statement

In a triangle ABCABC with circumcircle Γ\Gamma, MM is the midpoint of BCBC and point DD lies on segment MCMC. Point GG lies on ray BC→\overrightarrow{BC} past CC such that BCDC=BGGC\frac{BC}{DC} = \frac{BG}{GC}, and let NN be the midpoint of DGDG. The points PP, SS, and TT are defined as follows: [list = i] [*] Line CACA meets the circumcircle Γ1\Gamma_1 of triangle AGDAGD again at point PP. [*] Line PMPM meets Γ1\Gamma_1 again at SS. [*] Line PNPN meets the line through AA that is parallel to BCBC at QQ. Line CQCQ meets Γ\Gamma again at TT.
Prove that the points PP, SS, TT, and CC are concyclic.
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