What is the biggest area of a triangle can be drawn inside the ellipse equation :
For the circle , the largest triangle that can be drawn inside it is the equilateral triangle with side , whose area is .
The ellipse is obtained from the circle by the linear transformation with matrix . (In non-technical language, you get the ellipse from the circle by leaving the x-coordinate unchanged and expanding or shrinking the y-coordinate by a factor b/a.) This transformation alters areas by a factor b/a. So the largest triangle that can be inscribed in the ellipse has area , or square units when a=3 and b=7.
That method works fine provided that you assume that the base of the triangle is horizontal. What I found interesting (and unexpected) about this problem is that the same maximum area can be achieved with a triangle whose base has any given orientation. That follows easily if you think of the ellipse as a "squashed circle", but I think it would be hard to prove by any other method.