Problem

Source: Bundeswettbewerb Mathematik 2017, Round 2 - #3

Tags: geometry, incenter, geometry unsolved, Centroid, perpendicular



Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$. Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.