I just read about a theorem I had not heard of and thought I would share it with those who also have never seen it.
The theorem states that if we have a cubic polynomial with complex coefficients, and if the roots are 3 distinct non-collinear points A, B, and C in the complex plane, then the roots of f' are the foci of the unique ellipse inscribed in the triangle ABC and tangent to the sides at their midpoints.
To begin with, a cubic polynomial will have 3 roots in the complex plane.
We assume that those 3 roots are not all on a line, so they form the vertices of a triangle. Now, inside that trinagle, inscribe an ellipse. But let's make sure the ellipse is tangent to the sides of the triangle at their midpoints. This ellipse had two foci and those foci are the roots of the derivative of the cubic.
Here's a diagram to illustrate. The points where on the ellipse where the sides are tangent should be at the midpoints. I tried my best to draw it accordingly, but you get the idea. The foci represented by the w's are the roots of f' and the z's are the roots of the cubic.
Isn't it amazing how a cubic polynomial and an ellipse tie together?.
VERY cool. I tested it for a couple quick cubics, and sure enough, it works.
Where'd you hear about this? Does it have a title?
Most importantly, do you know of a proof?
Again, VERY cool!
Yes, it has a name....Marden's theorem.
A proof can be found in April's American Mathematical Monthly.
Also, visit here:
MathDL: Journal of Online Mathematics and its Applications
This has an interactive online presentation entitled "The most marvelous theorem in mathematics" by Dan Kalman.
That is impressive. (Yes) What I love about complex analysis is the very geometric nature. You can state many theorems in a very geometric way. There is actually an entire book written on the geometric side of complex analysis (though I never read it, people say it is good).