USAMO 1991 Problem 5

$\mathrm{X}$ is a point on the side $\mathrm{BC}$ of the triangle $\mathrm{ABC}$. Take the other common tangent (apart from $\mathrm{BC}$ ) to the incircles of ABX and ACX which intersects the segments AB and AC. Let it meet AX at Y. Show that the locus of $\mathrm{Y}$, as $\mathrm{X}$ varies, is the arc of a circle.