While it is easy to prove these facts analytically given the standard form equation for a general ellipse,
the proof never reveals why that they should be true. They were so much of a mystery that 200 years after Kepler, in the 19th century, a Belgian geometer named Germinal Dandelin gave a remarkable geometric proof of the focus-locus property of ellipses, using a construction that has come to be known as the Dandelin spheres.
The main picture of his construction is shown below, and you may learn more about his argument in Eric Weisstein's article at mathworld.
