The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman
What we will call the first uniqueness property for ellipses has the following statement.
The justification of this property is an immediate consequence of the Two Focus Property of ellipses. Given E, F, and P as in the statement, let d1 be the distance from E to P and let d2 be the distance from F to P. Then with d = d1 + d2, we see that the sum of the distances from P to points E and F equals d.
If P is not on the line segment joining E and F, then d is greater than the distance between E and F. In this case take E and F as foci of an ellipse with constant d, so that the ellipse is made up of all the points whose distances from E and F sum to d. This defines a unique ellipse, and P is a member of that ellipse. On the other hand, if P is on the line segment joining E and F, then d equals the distance from E to F. In this case, for any point Q, the sum of the distances to E and F equals d if and only if Q is on the line segment joining E and F, which is a degenerate ellipse. This establishes the First Uniqueness property.
To be consistent, we should distinguish between ellipses and degenerate ellipses. As defined elsewhere in these pages, the degenerate ellipse arises in a limiting case, and is not truly an ellipse. Therefore, when P is on the line segment joining E and F, it is technically incorrect to say that there is an ellipse through P with foci at E and F. This shows that the wording of the First Uniqueness Property is slightly inaccurate. This small abuse of notation should cause no confusion, and is committed for aesthetic reasons.