AUSTRIAN-POLISH 1981 Problem 3

Given is a triangle $ABC$ , the inscribed circle $G$ of which has radius $r$ . Let $r_a$ be the radius of the circle touching $AB$ , $AC$ and $G$ . [This circle lies inside triangle $ABC$ .] Define $r_b$ and $r_c$ similarly. Prove that $r_a + r_b + r_c \geq r$ and find all cases in which equality occurs.

Bosnia - Herzegovina Mathematical Olympiad 2002