  The law of centripetal force governing the motion of celestial bodies in
eccentric conic sections, has been established and thoroughly investigated by
Sir Isaac Newton in his Principia Mathematica. Yet its profound implications on
the understanding of such motions is still evolving. In a paper to the royal
academy of science, Sir Willian Hamilton demonstrated that this law underlies
the circular character of hodographs for Kepler orbits. A fact which was the
object of ulterior research and exploration by Richard Feynman and many other
authors [1]. In effect, a minute examination of the geometry of elliptic
trajectories, reveals interesting geometric properties and relations,
altogether, combined with the law of conservation of angular momentum lead
eventually, and without any recourse to dealing with differential equations, to
the appearance of the equation of the trajectory and to the derivation of the
equation of its corresponding hodograph. On this respect, and for the sake of
founding the approach on solid basis, I devised two mathematical theorems; one
concerning the existence of geometric means, and the other is related to
establishing the parametric equation of an off-center circle, altogether
compounded with other simple arguments ultimately give rise to the inverse
square law of force that governs the motion of bodies in elliptic trajectories,
as well as to the equation of their inherent circular hodographs.
