RMO 2014 M6

Let $D,E,F$ be the points of contact of the incircle of an acute-angled triangle $ABC$ with $BC,CA,AB$ respectively. Let $I_1,I_2,I_3$ be the incentres of the triangles $AFE, BDF, CED$ , respectively. Prove that the lines $I_1D, I_2E, I_3F$ are concurrent.