If the centroid of an equilateral triangle is $\left(-1,3\right)$  and one of its vertices is (5,0).  The equation of its circum circle is

In an equilateral triangle the distance between the circumcentre and any vertex is the radius of the circle

since centroid coincides with circumcentre

Radius, $r=AG=\sqrt{36+9}=\sqrt{45}$

 Equation of circle is ${\left(x+1\right)}^2+{\left(y-3\right)}^2=45$

$\Rightarrow x^2+y^2+2x-6y-35=0$