Dandelin spheres
What is this ?
This is to illustrate that a plane section of a cone is an ellipse because the sum of the distances to the two foci is constant.
Consider the two spheres that are tangent to the cone and to the plane. They touch the plane in two points thats appear to be the foci of the ellipse.
The reason why the coloured (green and cyan) segment lines are congruent is left to the reader. :-)
The diagram shows why their sum is constant.