# Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon.

This invariant is discussed briefly in the seminal paper On the Areas of Cyclic and Semicyclic Polygons (see section 5, equation (10)); a more in-depth discussion is had in a paper by Igor Pak Rigidity and polynomial invariants of convex polytopes (see sections 7.6 and 9); an elimination theory approach is discussed in a paper by Varfolomeev Inscribed polygons and Heron polynomials (the paper is not free to access). There are some more papers from the 2000's which deal with this problem, but the above papers seem to be the most encompassing.

I came across this problem when studying certain transformations of Chebyshev polynomials. I would like to go beyond the aforementioned papers, as in to give the "most explicit" formula for the aforementioned polynomial invariant in the general case, but this seems to be quite difficult.

I am asking if there have been any advances on finding a "most explicit" formula in the general case since 2004, and for other useful ideas.

## **Edit

By "general case" I mean the case where the number of sides of the polygon is arbitrary.

By "most explicit formula" I mean the monomial expansion.

• Could you clarify: What is "the general case"? Commented Jul 13, 2021 at 16:27
• @JosephO'Rourke When the number of sides of the polygon is arbitrary. There is the paper On Circumradius Equations of Cyclic Polygons where small cases are computed, but the formula for the general case leaves something to be desired in terms of "explicitness" Commented Jul 13, 2021 at 16:34