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(cevian) triangle area inequality with products, = iff AA',BB'CC' concurrent

Source: Sharygin 2006 X-XI CR 21

August 26, 2019

concurrencyconcurrentCevianareasarea of a trianglegeometric inequalitygeometry

Problem Statement

On the sides

AB, BC, CA

of triangle

ABC

, points

C', A', B'

are taken. Prove that for the areas of the corresponding triangles, the inequality holds:

S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}

and equality is achieved if and only if the lines

AA', BB', CC'

intersect at one point.