  The hodograph of the Kepler-Coulomb problem, that is, the path traced by its
velocity vector, is shown to be a circle and then it is used to investigate
other properties of the motion. We obtain the configuration space orbits of the
problem starting from initial conditions given using nothing more than the
methods of synthetic geometry so close to Newton's approach. The method works
with elliptic, parabolic and hyperbolic orbits; it can even be used to derive
Rutherford's relation from which the scattering cross section can be easily
evaluated. We think our discussion is both interesting and useful inasmuch as
it serves to relate the initial conditions with the corresponding trajectories
in a purely geometrical way uncovering in the process some seldom discussed
interesting connections.
