Show that the equidistant set of two points in R3 is a plane

Show that the equidistant set of two points in R^3 is a plane. Show also that the plane passes through O if the two points are both at distance 1 from O. Deduce from Exercise 7.4.1 that the equidistant set of two points on S^2 is a \"line\" (great circle) on S^. Next, we establish that there is a unique point on S^at given distances from three points not in a \"line.\" Suppose that two points P, Q S^2 have the same distances from three points A,S,C S^not in a \"line.\" Deduce from Exercise 7.4.2 that P = Q.

Solution

a) from 7.4.1 it is a plane passing through the origin in R^3 which is equidistant from two points on S^2. and intersection of this plane and S^2 is the line (great circle) in R^3.

b) from three points . there will be three planes . intersection of three planes will give a point.