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

Explanation

Allow `ΔABC` to be encircled by a circle of radius `r`. `AC` is depicted as a perpendicular `E` from the triangle's vertex `B`, which meets `AC` at `D` and circle at `E`.



Let `AC=2x` be the triangle's base.



and height of triangle `BD=y`



Then, area of triangle `A=1/2 xx 2x xx y` .......`(i)`


` {AC}` and `{BE}` are chords of circle.


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