A triangle inscribed in a circle has to be equilateral triangle for having maximum area ; how?
What if it is an ellipse instead of a circle, or a conic section in general ?

That would depend on the conic section.

Take a look at this: It is an ellipse inscribed in a cone.

https://www.askmehelpdesk.com/mathematics/volume-ellipsoid-inside-cone-365708.html


Find the ellipse of max area that can be inscribed in a circle of radius R?