I’ll also mention that this is exactly the geometric proof I referred to. The one for hyperbolas is similar. However, I can’t think of one that works for parabolas.

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That should have appeared after paragraph 7, but it looks like it was filtered out.

]]>“Ross Honsberger has spent more than two decades collecting mathematical morsels for general computation. Attending one of his rare public lectures, I had the pleasure of sampling one of his mathematical feasts. I found it not just palatable but downright delicious.

“I listened to Honsberger’s lecture near my hometown in Ontario. He served to the audience such delights as spheres in a cone, checkers on a board, dots on a dish and beans in a Greek urn.

“Honsberger began by describing the marvelous spheres of Germinal Dandelin, a 19th-century Belgian mathematician. Dandelin discovered an amazing connection between the classical and modern concepts of the ellipse. The Greeks conceived of an ellipse as the figure that results when a plane cuts obliquely through a cone. Since the time of Descartes, however, the ellipse has been described analytically in terms of two special points called foci. The sum of the distances from the two foci to any point on the ellipse is constant.

“Honsberger introduced Dandelin’s spheres by drawing our attention to a projected transparency of a plane cutting a cone. (Readers can follow Dandelin’s argument with occasional glances at Figure 13.1.) I cannot swear that what follows are Honsberger’s exact words, but he readily admits to a certain broad similarity:

“‘It takes no genius to see that the plane divides the cone into two pieces. But it was Dandelin’s idea to insert a sphere into each piece. Like an over-inflated balloon. each sphere contacts the wall of the cone and touches the elliptical plane at a certain point. But where? One can imagine Dandelin’s heart leaping at the thought that the spheres might touch the plane at the two focal points of the ellipse.’

“Honsberger places his marking pen on the transparency. He labels the two points of contact by the symbols F and G. Are these the foci of the ellipse?

“‘Let’s take a look at what clever old Dandelin did. First, through any point P that we care to select on the ellipse, we may draw a straight line that runs up the side of the code to its tip. Second, the line will touch the two spheres at two points, say, A and B. No matter where we pick P to lie on the ellipse, the length of AB will be the same.

“‘Ah, but that gives it away! The distance from the point F to P equals the distance from P to A. After all, both PF and PA are tangents to the same sphere from the same point. By the same reasoning, the distance from the second point, G, to P equals the distance from B to P. Are we not finished? PF PG = PA PB, and the latter sum is just the (constant) length of AB.

“‘Now isn’t that the darndest thing?’ Honsberger sounds rural.

“As I look around the lecture hall, students appear stunned. Professors alternately smile and frown. One of them behind me murmurs, ‘Well, I’ll be.’”

A.K.Dewdney ran the Computer Recreations and Mathematical Recreations columns in Scientific American for many years. His columns (and more) appear reprinted in his books The Tinkertoy Computer, The Magic Machine, The Armchair Universe, and The Turing Omnibus. All are fascinating reads.

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