KöMaL 2017 B4919

Let $I$ be the centre of the inscribed circle of triangle $ABC$. The inscribed circle touches the sides $BC$, $CA$ and $AB$ at the points $A_1$, $B_1$ and $C_1$, respectively. The lines $AB$ and $A_1B_1$ intersect at point $K$. Show that the circle centred at $K$ and passing through $C_1$, and the inscribed circles of the kites $AC_1IB_1$ and $BA_1IC_1$ have a tangent in common, and that tangent passes through the point $C$.