Question 79
The incircle of a triangle $ABC$ is tangent to $BC$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$. The $B$-mixtilinear incircle, centered at $O_{B}$, is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_{C}$, is defined similarly. Suppose that $\overline{DH} \perp \overline{O_{B}O_{C}}$, $AB=\sqrt{3}$ and $AC=2$. Find $BC$.