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triangle in ellipse

brian bovril 894
December 18 2006
0
Does anyone know how to work out the largest triangle that can be enclosed within an ellipse (both in 2D)? For any triangle, Area:=sqrt(s*(s-a)*(s-b)*(s-c)) where s:=(a+b+c)/2 and a,b,c are the length of the sides. Say the ellipse is equation 1/9*x^2+1/4*y^2 = 1 hence y: = -2/3*(-x^2+9)^(1/2) or +2/3*(-x^2+9)^(1/2) Given three points (x1,y1), (x2,y2), (x3,y3) Eg distance from (-x1,-y) to (-x2,y) with(student): a:=distance([-x1, subs(x=-x1,-2/3*(-x^2+9)^(1/2) )], [-x2, subs(x=-x2, +2/3*(-x^2+9)^(1/2))]); (note the sign of y) and so forth for b and c. there has to be a more eloquent (calculus) way.
calculus

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