If the incentre of an equilateral triangle is 1 , 1 and the equation of its one side is 3 x + 4 y + 3 = 0 ,…

$3x+4y+3=0$

Since, for equilateral triangle centroid in center and circumcenter is the same therefore length of $AB=\left|\frac{3\left(1\right)+4\left(1\right)+3}{\sqrt{3^2+4^2}}\right|$

$=\frac{10}{5}=2$

$R=2\left(AB\right)=4$

As centroid divide median in the ratio $2:1$

Centre of the circle would be $\left(1,1\right)$

$\therefore$ Answer is $x^2+y^2-2x-2y-14=0$