Nemotron-IMO-Bench 2026 G35

Let $D$ be a point on the side $BC$ of triangle $ABC$ and let $\Gamma$ be the circumcircle of triangle $ABC$. Let $\mathcal{B}, \mathcal{C}$ be the circles tangent to $\Gamma$, $AD$, $BD$ and to $\Gamma$, $AD$, $CD$ respectively. Prove that $\mathcal{B}$ is tangent to $\mathcal{C}$ if and only if the angles $BAD$ and $CAD$ are equal.