Let E be the ellipse of maximum area that can be inscribed in a triangle of side lengths 1, √3, 2.
Find the distance between the foci of the ellipse E.
The ellipse described in the problem is the
Steiner Inellipse of the triangle. That page gives formulas for the key dimensions of the ellipse in terms of the sides of the triangle (sa,sb,sc).
Z = sqrt(sa^4 + sb^4 + sc^4 - sa^2*sb^2 - sa^2*sc^2 - sb^2*sc^2) = sqrt(1 + 9 + 16 - 3 - 4 - 12) = sqrt(7)
semimajor axis a = (1/6)*sqrt(sa^2 + sb^2 + sc^2 + 2*Z) = (1/6)*(sqrt(1 + 3 + 4 + 2*sqrt(7)) = (1/6)*sqrt(8+2*sqrt(7))
semiminor axis b = (1/6)*sqrt(sa^2 + sb^2 + sc^2 - 2*Z) = (1/6)*(sqrt(1 + 3 + 4 - 2*sqrt(7)) = (1/6)*sqrt(8-2*sqrt(7))
focal length c = (1/3)*sqrt(Z) = (1/3)*7^(1/4)
Then the distance between the foci of the ellipse is twice the focal length: (2/3)*7^(1/4).
Easy answer
