Trigonometry! #55

Notation: $r_{1}$ , $r_{2}$ and $r_{3}$ are the exradii, $s$ is the semiperimeter, $\Delta$ is the area and $a$ , $b$ and $c$ are the sides of the triangle.

As the exradii are in a harmonic progression, we have $\frac{2}{r_{1}}=\frac{1}{r_{1}}+\frac{1}{r_{3}}$

$\frac{2(s-b)}{\Delta}=\frac{s-a}{\Delta}+\frac{s-c}{\Delta}$

$2(s-b)=s-a+s-c$

$2s-2b=2s-a-c$

$\boxed{2b=a+c}$

$\therefore$ The sides are in an arithmetic progression.