KöMaL 2007 B3979

The vertices of triangle ABC are labelled in counterclockwise order. The angles at the vertices $A$, $B$, and $C$ are $\alpha$, $\beta$ and $\gamma$, respectively. The vertex $B$ is rotated about the point $A$ through the angle $\alpha$ in clockwise direction. The point $B_1$ obtained is rotated about $B$ through the angle $\beta$ in clockwise direction. Finally, the point $B_2$ obtained is rotated about $C$ in clockwise direction to the point $B_3$. Given the points $B$, $B_3$ and $O$ the middle point of the incentre of the triangle ABC , construct the triangle. Investigate how the solution depends on the positions of the three given points.