On Geometric Property of Fermat-Torricelli Points on Sphere

Zhenbing Zeng, Yu Chen, Xiang Sun, Yuzheng Wang

Abstract: Given three points on sphere \(S^2\), a point on sphere that maximizes or minimizes the sum of its Euclidean distances to the given points is called Fermat–Torricelli point. It was proved that for \(A,B,C\in S^2\) and their Fermat–Torricelli point P, the distance sum \(L=PA+PB+PC\) and the edges \(a=BC, b=CA, c=AB\) satisfy a polynomial equation \(f(L,a,b,c)=0\) of degree 12. But little is known about the geometric property of Fermat–Torricelli points, even when A, B, C are on very special positions on sphere. In this paper, we will show that for three points A, B, C on a greater circle on sphere, their Fermat–Torricelli points are either on the same greater circle or on one of four special positions (called Zeng Points) determined by A, B, C.