Belarusian Mathematical Olympiad

Points $B$ and $C$ are marked on the half-hyperbola $y=1/x$ which lies in the first quadrant of the Cartesian plane. The abscissa of $C$ is greater than the abscissa of $B$. Let $A$ be the intersection point of the other half-hyperbola and the line passing through the origin and $B$.

Prove that the angle $BAC$ is equal to one of the angles between the line $BC$ and the tangent to the hyperbola at point $B$.