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

Yet another approach to the inverse square law and to the circular character of the hodograph of Kepler orbits