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cevian inequality

Source: S&M 2002 1st Grade P2

May 14, 2021
Geometric Inequalitiesgeometric inequalityinequalitiesgeometry

Problem Statement

Let OO be a point inside a triangle ABCABC and let the lines AO,BOAO,BO, and COCO meet sides BC,CABC,CA, and ABAB at points A1,B1A_1,B_1, and C1C_1, respectively. If AA1AA_1 is the longest among the segments AA1,BB1,CC1AA_1,BB_1,CC_1, prove that OA1+OB1+OC1≤AA1.OA_1+OB_1+OC_1\le AA_1.
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