General test to prove a set of points creates a cyclic polygon

Question

For $n>4$, is there a way to show that a set of points creates an $n$-sided, cyclic polygon? Or simply, that the set of points are concyclic?

All triangles are cyclic. For quadrilaterals, we can use Ptolemy's theorem about the product of diagonals, among others. But, can these tests be extended to arbitrary sided polygons?

The Japanese theorem for cyclic polygons is one good answer, but I am wondering if there are other simpler ways.

Just check that for each $k$, $P_1P_2P_3P_k$ is concyclic, by say Ptolemy. — Angina Seng, Jul 15 '18 at 08:37
It's probably good to note that Lord Shark's idea works because of Caratheodory's theorem. — Theo Bendit, Jul 15 '18 at 08:41
@LordSharktheUnknown I was expecting to use all the points simultaneously, but that's a very good point too. — John Glenn, Jul 15 '18 at 08:46

score 1 · Answer 1 · answered Jul 15 '18 at 11:23

An alternative is the following. Take any three points $p_1, p_2, p_3$ and compute the circle $C$ they determine:

Then check each of the subsequent points, $p_4, p_5, \ldots$ to lie on $C$. You may need to take care with numerical issues.

For how to compute a $3$-point circle, see, e.g., Ed Pegg's Mathematica Demo.

General test to prove a set of points creates a cyclic polygon

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