  [This is an expository article. I have submitted it to the American
Mathematical Monthly.] The three-body problem defines a dynamics on the space
of triangles in the plane. The shape sphere is the moduli space of oriented
similarity classes of planar triangles and lies inside shape space, a Euclidean
3-space parametrizing oriented congruence classes of triangles. We derive and
investigate the geometry and dynamics induced on these spaces by the three-body
problem. We present two theorems concerning the three-body problem whose
discovery was made through the shape space perspective
