On the Uniqueness of Co-circular Four Body Central Configurations

On the uniqueness of trapezoidal four-body central configurations

“…Proof. The proof is analogous to the proof of Lemma 6 in [29], and to Smale's proof of Moulton's theorem for the collinear n-body problem [31] (which however, is presented without details). We repeat it here for convenience of the reader.…”

Some Polynomial Conditions of Cyclic Quadrilaterals, Tilted Kites and Other Quadrilaterals

“…Since the points are on the plane we also have that CM = 0. Hence equation (11) gives that K = 0. The equation K = 0 can also be written as…”

“…The first and third equations above provide interesting way to organize the terms of the Cayley-Menger determinant. The first of these identities has been useful in studying cyclic central configurations in Celestial Mechanics [2,11]. We begin by stating without proof the well known Ptolemy's theorem:…”

“…The polynomials studied by Pech [9] have been useful in studying cyclic central configurations of four bodies in Celestial Mechanics [2,11]. Similar polynomials, related to another type of quadrilaterals, namely trapezoids, have also been found to be useful in studying central configurations of four bodies [10,12].…”

“…These polynomials relationships are very similar to the ones found by Pech [9], and therefore it is not worthwhile to study them. However, combining these equations with (10) and (11) we can obtain the following symmetrized relations…”