What are the possible values of $\angle CAB$?

Question

In an acute $\triangle ABC$, the segment $CD$ is an altitude and H is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the $\angle DHB$, determine all possible values of $\angle CAB$ .

Answer 1

Observe that the exterior bisector of $\angle BHC$ meets the perpendicular bisector of $BC$ in the midpoint of the arc $BHC$ of circle $BHC$ so $BHOC$ is concyclic [$O$ is the circumcentre] so $180-\angle BAC = 2\times \angle BAC$ so $\angle BAC=60^{\circ}$.