The proof of the three reflections theorem begins, as it did for R², by considering the equidistant set of two points. 7.4.1 Show that the equidistant set of two points in R³ is a plane. Show also that the plane passes through O if the two points are both at distance 1 from O. 7.4.2 Deduce from Exercise 7.4.1 that the equidistant set of two points on S² is a "line" (great circle) on S². Next, we establish that there is a unique point on S2 at given distances from three points not in a "line." 7.4.3 Suppose that two points P,QE S² have the same distances from three points A,B,C S² not in a "line." Deduce from Exercise 7.4.2 that P = Q. 7.4.4 Deduce from Exercise 7.4.3 that an isometry of S² is determined by the images of three points A, B, C not in a "line." Thus, it remains to show the following. Any three points A,B,C E S2 not in a "line" can be mapped to any other three points A', B', C'ES², which are separated by the same respective distances, by one, two, or three reflections. 7.4.5 Complete this proof of the three reflections theorem by imitating the argu- ment in Section 3.7.