Problem

Source: Romanian TST 1997

Tags: geometry, circumcircle, incenter, geometric transformation, homothety, ratio, power of a point



Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. Dan Branzei