On a property of the Vesica Piscis

Vesica Piscis is a figure generated intersecting two circles having the same radius and such that the center of each circle lies on the exterior circumference of the other circle. The origin of the Vesica Piscis is ancient and unknown and this symbol has been present in all the ages up to nowadays. It has been proved that if two segments are traced joining particular points of the Vesica Piscis, these segments form a cross and their ratio is equal to the square root of three. One segment joins the two intersecting points of the two circumferences and the other segment joins the two intersecting points between the segment joining the two centers with the exterior circumferences themselves. In this article, we prove that the converse is also true: if we consider two intersecting circles and the aforementioned segments such that their ratio is equal to the square root of three, then the two circles form a Vesica Piscis. We prove this results through two alternative proofs. The first proof uses Euclidean geometry and the second proof uses analytic geometry.