TITLE: Minimal Ellipse Circumscribing A Right Triangle
QUESTION [10 upvotes]: Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one.
You may chose the origin and orientation of the $x,y$ axes as you want.
Motivation: It can be proved [Problem of the Week, Problem No. 8 (Fall 2008 Series), Department of Mathematics, Purdue University] that the area of this ellipse is $8\pi /\sqrt{3}$, without the need of using its equation, but I am also interested in finding it.

Edit: picture from this answer.

REPLY [2 votes]: Not much to contribute at this point except this enhanced picture:

(dashed lines: axes; gray lines: medians)
and the Mathematica notebook used to produce it.
(Thanks Isaac!)