Given a Circle $B_r$ and a triangle $\triangle ABC$ in it with two fixed (immovable) points $A$ and $B$. The third point $C$ can be moved on the circle. I want to prove the following equivalence: (for a visualization see picture below)
The area of the triangle $\triangle ABC$ is extremal (with regard to the position of $C$) if and only if the sum of the lengths $|AC|+|CB|$ is extremal (with regard to the position of $C$).
I know already that this is the case if the Point $C$ is chosen to be the point at which the Tangent $T_c B_r$ is parallel to $|AB|$ or, equally, $\alpha=\beta$.