Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.

Explanation

OB=9
BD=`sqrt(a^2-x^2)`
CD=`sqrt(a^2-x^2)`
` AD=a+x`
Area=`1/2*(a+x)*2sqrt(a^2-x^2`
A=`(a+x)sqrt(a^2-x^2`
`(dA)/(dx)=sqrt(a^2-x^2)+(a+x)/(2sqrt(a^2-x^2)`-2x=0.
`=2a^2-2x^2-2ax-2x^2=0`
`a^2-ax-2x^2=0`
`(a-2x)(a+x)=0`
`a-2x=0 or a+x=0`
`x=a/2,-x`
BD=`sqrt(a^2-x^2)=sqrt3/2a`
BC=`sqrt3a`
`AB^2=3/4a^2+9/4a^2=3a^2`
`AB=sqrt3a`.
BC and AB are equal.