hii

 

Now since the circle is the cirumcircle of the triangle we know the following results ..

 

well a = 2rsinA etc

 

Also area of triangle is 1/2. ab sinC

 

                         =  1/2. (2r sinA)(2r sinB)sinC

                         =  2.r² sinA.sinB.sinC

 

so now to maximize the area we need to maximize sinA.sinB.sinC 

 

Now sinA.sinB.sinC = sinA.sinB.sin(A+B)

 

To maximize we differentiate w.r.t A and put it zero ..

 

so sinB.cos(A+B) + cosB.sin(A+B) = 0

 

or sin(A+2B) = 0

 

or A + 2B = 180

 

but we know A + B + C = 180

 

so,  B = C

 

By symetry we get A = B = C = 60

 

and hence the triangle is equilateral ..

 

cheers