ILL 1984 Problem 48

Let $ABC$ be a triangle with interior angle bisectors $AA_1, BB_1, CC_1$ and incenter $I$ . If $\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC]$ , where $\sigma[ABC]$ denotes the area of $ABC$ , show that $ABC$ is isosceles.