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! At 03:17 PM 2/18/2013, James Cloos wrote:
"HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> Where do the Marden ellipse foci go when the 3 vertices are collinear? HB> Suppose A,B,C are the 3 triangle vertices, with their opposite line segments a,b,c, respectively, as usual.
HB> Suppose we go to the limit where a=b+c. As the triangle collapses, HB> the foci move further and further into the corners B,C. In the HB> limiting case, the foci are _at_ B,C.
That was my intuition (or memory? I haven't really thought about such stuff much in the last 25 or so). But a simple poly with real (and therefore collinear) zeros such as 0=(x-1)*(x-2)*(x-3) seems to show otherwise.
The roots of the 1st diff are bounded by 2+-sqrt(3)/3, and the root of the 2nd diff is 2, (as calculated and plotted by maxima).
Or does this imply the that limit of an ellipse inscribed in a triangle is not the same as the limit of said triangle when the triangle is collapsed to a line segment?
-JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6