Školjka

%V0 Two circles $\Omega_{1}$ and $\Omega_{2}$ are externally tangent to each other at a point $I$, and both of these circles are tangent to a third circle $\Omega$ which encloses the two circles $\Omega_{1}$ and $\Omega_{2}$. The common tangent to the two circles $\Omega_{1}$ and $\Omega_{2}$ at the point $I$ meets the circle $\Omega$ at a point $A$. One common tangent to the circles $\Omega_{1}$ and $\Omega_{2}$ which doesn't pass through $I$ meets the circle $\Omega$ at the points $B$ and $C$ such that the points $A$ and $I$ lie on the same side of the line $BC$. Prove that the point $I$ is the incenter of triangle $ABC$. Alternative formulation. Two circles touch externally at a point $I$. The two circles lie inside a large circle and both touch it. The chord $BC$ of the large circle touches both smaller circles (not at $I$). The common tangent to the two smaller circles at the point $I$ meets the large circle at a point $A$, where the points $A$ and $I$ are on the same side of the chord $BC$. Show that the point $I$ is the incenter of triangle $ABC$.