RO-TST 2008 Problem 18

Let $ABC$ be a triangle and let $\mathcal{M}_{a}$ , $\mathcal{M}_{b}$ , $\mathcal{M}_{c}$ be the circles having as diameters the medians $m_{a}$ , $m_{b}$ , $m_{c}$ of triangle $ABC$ , respectively. If two of these three circles are tangent to the incircle of $ABC$ , prove that the third is tangent as well.