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Geometric inequality with areas

Source: Czech and Slovak Olympiad 2015, National Round, Problem 5

April 1, 2015
geometrygeometric inequalityincenter

Problem Statement

In given triangle △ABC\triangle ABC, difference between sizes of each pair of sides is at least d>0d>0. Let GG and II be the centroid and incenter of △ABC\triangle ABC and rr be its inradius. Show that [AIG]+[BIG]+[CIG]≥23dr,[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr, where [XYZ][XYZ] is (nonnegative) area of triangle △XYZ\triangle XYZ.
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