This Demonstration considers the intersection of a cone with a plane. This conic section is shown to be an ellipse. Two Dandelin spheres are drawn, inscribed in the cone above and below the plane. The two foci of the ellipse are the points of tangency of the Dandelin spheres with the plane, shown as black points.
The tangency of the blue line to the two spheres implies that the sum of the lengths of the black and red lines is equal to the sum of the lengths of the two blue line segments joined by the three red points. This is a defining property of an ellipse: the locus of points such that the sum of the distances to the two foci is constant.