I just ran across this "Most Marvelous Theorem in Mathematics": http://www.maa.org/joma/Volume8/Kalman/index.html It is well-known/well-taught that the roots of p'(x) lie within the convex hull of the roots of p(x). However, in the case of a cubic, we can say a lot more: the roots of p'(x) are the _foci_ of the inscribed _ellipse_ that passes through the midpoints of the sides of the triangle formed by the roots of p(x). This is Marden's/Siebeck's theorem. So in the case of (my) standardized cubic p(x)=x^3-(3/4)*x-cos(3*a)/4 [roots are cos(a), cos(a+2*pi/3), cos(a-2*pi/3), even when a is complex], the roots of p'(x) are +-1/2. So even as the triangle becomes enormous (a is complex), the ellipse becomes closer & closer to a circle, and the triangle becomes closer & closer to an equilateral triangle. Marden's/Siebeck's Theorem is one property of ellipses that Newton was almost certainly not aware of. Are there any consequences of this theorem that show up in Newton's 2-body problem? Given an ellipse, is the triangle that the ellipse is inscribed within unique? If so, are there any interesting physical/mechanical consequences of those triangle vertices? My conjecture is that this inscribed ellipse may describe an maximum speed trajectory that lies within the triangle, subject to certain acceleration constraints, but I don't know enough calculus of variations to be able to get any further.