A simple derivation of the two force laws for elliptic orbits from Proposition 6 in Newton's Principia

Based on Propostion 6 of his Principia, Newton’s geometrical derivation in Propositions 10 and 11 for the radial dependence of the two central forces that lead to elliptical orbits is notoriously difficult. An alternate and more transparent derivation is obtained by applying the affine transformation of a circle into an ellipse.