Feynman's proof of the law of ellipses

Kepler’s 1st law arises from the 2nd and 3rd Kepler’s laws and from the fact that the gravitational force diminishes as R-2 (the inverse square ofthe distance law of gravity). Feynman proved that to equal anglescorrespond equal velocity changes. He illustrated this property by two diagrams: an orbit diagram and a velocity diagram. From the construction (and from the relevant physical laws) follows that the velocity diagram is always of the shape of the regular polygon
Feynman found the geometrical correspondence of these two diagrams based on the equality of the angle swept by a planet with the relevant central angle in the velocity diagram, see the figure below.  
Rotating the velocity diagramand using the properties of ellipse that we mentioned above he proved that the shape of the planet’s path is an ellipse.