(I) Determine the maximum area of an ellipse that is inscribed within an integer triangle having sides 13, 14, and 15.
(II) Determine the minimum area of an ellipse that is circumscribed around an integer triangle having sides 13, 14, and 15.
This problem is nearly trivial if you know that the two ellipses described are the Steiner inellipse (part 1) and Steiner circumellipse (part 2). For any triangle the areas of these ellipses compared to the triangle is
4*pi : 3*sqrt(3) : pi <=> area(circumellipse) : area(triangle) : area(inellipse)
By Heron's formula the area of the triangle is 84. Then for part 1 that ellipse's area is 84*(4*pi)/(3*sqrt(3)) = 112*pi/sqrt(3) = 203.15. And then for part 2 the area of that ellipse is 84*(pi)/(3*sqrt(3)) = 28*pi/sqrt(3) = 50.79.
We had a problem very similar to this many years ago:
Ellipses.
Solution
