I've seen the proof, using 2 inscribed spheres, that when a plane intersects a cone in a way that the intersection is a closed curve, the intersection will be an ellipse.

But aren't there other demonstations of that fact? Is there a briefer proof, or one that is naturally and obviously motivated? Of course analytic geometry could prove it too, but that would be a long brute-force demonstration.

The reason why I ask is that sometimes things can be demonstrated in ways that are brief, and naturally motivated. For instance, there was an article in _Scientific American_ showing several brief, simple, and natural demonstrations of the Pythagorean theorem, demonstrations much simpler than the standard one.

Another request:

How about a brief and natural demonstration that when lines are drawn from a point to all the points on a circle, the surface formed by the lines is an elliptical cone. (If the circle is perpendicular to the line connecting its center to the point,then of course the elliptical cone is a circular cone).

Mike Ossipoff