What is the area of the smallest ellipse that can be circumscribed around a 3-4-5 triangle?
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What is the area of the largest ellipse that can be inscribed in a 3-4-5 triangle?

The area ratio of a triangle to a circumscribed, or inscribed, ellipse remains constant under affine transformations (combinations of linear transformations and translations). Under affine transformations, triangles go to triangles and ellipses go to ellipses. Any two triangles can be transformed into each other by affine transformation. For an equilateral triangle, the circumscribed ellipse of minimum area, and the inscribed ellipse of maximum area, are the circumscribed and inscribed circles, respectively. The ratios of circle area to triangle area are thus 4*pi/3*sqrt(3)=2.418399..., and pi/3*sqrt(3)=0.6044599... since the equilateral triangle with side sqrt(3) has the circle of unit radius as its circumscribing circle and the circle of radius 1/2 as its inscribed circle. The area of the given 3,4,5-triangle is 6, so its circumscribed ellipse of minimum area has area 24*pi/3*sqrt(3)=8*pi/sqrt(3)=14.510... and its inscribed ellipse of maximum area has area 2*pi/sqrt(3)=3.62759... .

This is the same result as Brian Smith has just given below, except that it is confined just to the areas, which were the only quantities sought by the problem statement.

Posted by Richard on 2004-05-18 14:54:06