This is not what happens. The limit of the ellipse as c -> 0 is indeed a degenerate ellipse that is precisely the line segment between its foci, +-sqrt(⅓). --Dan P.S. Could we *please* given credit where credit is due, and call the theorem in question "Siebeck's Theorem", since it was discovered 149 years ago by someone named Siebeck? For unknown reasons, Dan Kalman, in his 2008 article in the Monthly, decided to call it Marden's Theorem. (There were indeed articles by one Morris Marden about the theorem: one in 1945, and another in 1966.) But the theorem was discovered by Jörg Siebeck in 1864. Kalman acknowledges this in his article. See the unfortunately named Wikipedia article: < http://en.wikipedia.org/wiki/Marden's_theorem >. On 2013-02-18, at 5:19 PM, Henry Baker wrote:
Re Marden's Theorem:
Interesting!
Consider the polynomial P(z)=(z-1)*(z+1)*(z-c*%i), which describes an isosceles triangle sitting on the real number line from -1 to +1 with altitude c.
If I compute P'(z) and find its rightmost zero, I get
2 sqrt(3 - c ) + %i c (%o18) z = ------------------- 3
whose real part is 2 sqrt(3 - c ) (%o21) ------------ 3
so long as |c|<=sqrt(3).
As c->0, this real part ->1/sqrt(3).
So the foci of the ellipse _don't_ approach the corners after all.
I hadn't thought it possible that an ellipse could approach a line segment while still keeping its foci away from the line segment ends, but I was wrong!