Details

The Demonstration works as follows: The polynomial is differentiated and the roots of its derivatives are plotted as orange points. The ellipse with these points as its foci and passing through the midpoint of one side of the triangle is then plotted. The fact that this ellipse is inscribed in the triangle, with points of tangency being the midpoints of all three sides, illustrates the theorem.

For details see Dan Kalman's article, "The Most Marvelous Theorem in Mathematics", in Journal of Online Mathematics and Its Applications. Further references can be found there.