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`.