  We carry out a sequence of coordinate changes for the planar three-body
problem which successively eliminate the translation and rotation symmetries,
regularize all three double collision singularities and blow-up the triple
collision. Parametrizing the configurations by the three relative position
vectors maintains the symmetry among the masses and simplifies the
regularization of binary collisions. Using size and shape coordinates
facilitates the reduction by rotations and the blow-up of triple collision
while emphasizing the role of the shape sphere. By using homogeneous
coordinates to describe Hamiltonian systems whose configurations spaces are
spheres or projective spaces, we are able to take a modern, global approach to
these familiar problems. We also show how to obtain the reduced and regularized
differential equations in several convenient local coordinates systems.
