# Math Help - Triangle

1. ## Triangle

Prove that if the line through the centroid and the incenter is parallel to a side then the sides of the triangle form an arithmetic sequence.

2. Suppose that $IG\parallel BC$ (I=incenter, G=centroid).
Let $r$ be the inradius, $h_a$ the height from the vertex A, $S$ the area of triangle and $p$ the semiperimeter.

Then we have $r=\frac{h_a}{3}=\frac{2S}{3a}$

But $r=\frac{S}{p}\Rightarrow\frac{2S}{3a}=\frac{S}{p}\ Rightarrow\frac{1}{3a}=\frac{1}{a+b+c}\Rightarrow a=\frac{b+c}{2}$ and that means that b, a, c (or c, a, b) are in arithmetic progression.