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AG + BG + CG <= A_1C + B_1C + C_1C, centroid , circumcircle

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2000 Seniors p4

September 6, 2020
geometryCentroidgeometric inequalitycircumcircleChampions Tournament

Problem Statement

Let GG be the point of intersection of the medians in the triangle ABCABC. Let us denote A1,B1,C1A_1, B_1, C_1 the second points of intersection of lines AG,BG,CGAG, BG, CG with the circle circumscribed around the triangle. Prove that AG+BG+CG≤A1C+B1C+C1CAG + BG + CG \le A_1C + B_1C + C_1C.
(Yasinsky V.A.)
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