The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman
What we will call the second uniqueness property for ellipses has the following statement.
Let E and F be distinct points in the plane, and let L be any line that does not intersect the closed line segment joining E and F. Then there exists a unique ellipse with E and F as foci and which is tangent to L.
To justify this property we will consider the family of ellipses which all have E and F as foci. Each ellipse can be defined by a particular choice of the constant d, as specified in the Two Focus Definition for ellipses. The value of d must be at least as great as the distance between E and F. The larger the value of d, the larger the ellipse, the smaller the value of d, the smaller the ellipse, and the limiting degenerate case, with d equal to the distance between E and F, is just the segment joining E and F. Several members of the family of ellipses are shown in Figure 1.
By taking d close enough to the distance between E and F, we can produce an ellipse arbitrarily close to the segment from E to F. In particular, for an appropriate choice of d, the ellipse will be disjoint from line L. This is valid because there is a definite least distance r between any point of the line segment and the line L. Indeed, r can be taken as the distance from line L to E or F, whichever is closer -- from every other point of the line segment, the distance to line L is greater than or equal to r. Now for d close enough to the distance between E and F, every point of the corresponding ellipse will be within r/2 of the line segment. Therefore, no point of that ellipse can lie on L. This shows that there will always exist ellipses like the blue ones in the figure.
On the other hand, as d increases, its ellipse expands outward without bound. Therefore, there must also be ellipses like the red ones in the figure, which intersect line L. The transition from the blue to the red ellipses corresponds to an ellipse that is tangent to line L.
To make this argument more precise, consider the set D of values of d for which the corresponding ellipse is disjoint from L . We know that there are such (blue) ellipses, so D is not empty. And the existence of the red ellipses shows that set D has an upper bound. This implies that set D has a least upper bound, d*. The ellipse for d* is the unique ellipse in the family that is tangent to L.
This is illustrated in Figure 1. The line L is shown in green. The smallest values of d correspond to the blue ellipses that are disjoint from L. The largest values of d are for the red ellipses which each intersect L in two points. The green ellipse corresponds to d = d*. For this value of d, the ellipse just touches L, and so is tangent to L.