Geometrically interpreting complex numbers. Prove that
$|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$
where $0 < \alpha < \beta < \gamma < \delta < 2\pi $
and interpret this result as a geometric theorem.
I found this question on a university entrance exam past paper. The question quite interesting and while it is not too hard to prove using a 'nuts and bolts' approach I was wondering if there was an elegant approach using less trigonometry.
Thanks.

One can notice that if we introduce four points associated with $\alpha, \beta, \gamma$, and $\delta$, then these points are lying on a unit circle. Moreover, since $\alpha<\beta<\gamma<\delta < 2\pi$, these points form a non-intersecting polygon, for which Ptolemy's theorem (https://en.wikipedia.org/wiki/Ptolemy%27s_theorem) applies.
