You
might like to press F11 to view the Microworld in Full Screen
mode.
Marden's
theorem concerns polynomials over the complex numbers.
As
a specific case, consider the polynomial p(z) = (z-a)(z-b)(z-c),
where a, b, and c are fixed complex numbers, and z is a
complex variable. We know that p has roots at a, b, and
c, which we visualize as three points in the complex plane.
If those points are not colinear, they define a triangle
within which it is possible to inscribe an ellipse, tangent
to the sides at their midpoints.
Marden's
theorem says that the foci of the ellipse are precisely
the roots of the derivative, p'(z).
In
this Microworld, you may experiment with this, and a more
general version of Marden's Theorem. And you may review
properties of complex numbers and the complex plane, both
visually and algebraically. This Microworld will also give
you the opportunity to explore the geometry of ellipses
inscribed in triangles. This is a "hands on" opportunity
to investigate some beautiful correlations between algebra
and geometry. At every step, the book invites you to ask
questions, and to see for yourself what the answers to those
questions are. You will enjoy this one!
Author:
Dan Kalman, Project WELCOME Co-PI
E-mail: kalman@american.edu
