Authorea

In this case, \(Q=Q^{\prime}\). Observe the angles \(\angle PQ^{\prime}A\) and \(\angle PQ^{\prime}B\). WLOG assume \(\angle PQ^{\prime}B\) is the larger of the two and denote it \(\alpha\).

\begin{align} \angle PQ^{\prime}A+\angle PQ^{\prime}B & =\pi\notag \\ \therefore\alpha:=\max\left(PQ^{\prime}A,PQ^{\prime}B\right) & \geq\frac{\pi}{2}\notag \\ \end{align}

As such \(\alpha\) must be the unique largest internal angle in the triangle \(PQ^{\prime}B\). Consequently the side opposite \(\alpha\) \(\left(PB\right)\) is the unique hypotenuse of the triangle. As such \(|PB|>|PQ^{\prime}|\). However \(P\) and \(B\) are both elements of the convex hull so we have found a line segment with length larger than our assumed maximum, a contradiction.