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MI = r/3 iff MI _|_ to a side, where M,I centroid , incenter of scalene

Source: 2012 Sharygin Geometry Olympiad Final Round 10.3

August 3, 2018

geometryCentroidincenterperpendicular

Problem Statement

Let

M

and

I

be the centroid and the incenter of a scalene triangle

ABC

, and let

r

be its inradius. Prove that

MI = r/3

if and only if

MI

is perpendicular to one of the sides of the triangle.
(A.Karlyuchenko)