The equation of the circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a  is

As the centre of the circumcircle of an equilateral triangle is its Centroid and the distance of the Centroid from the vertex of the triangle is 2/3 of its medium through the vertex. So the distance of the centre from a vertex of the triangle is (2/3)×3a=2a  and hence the vertex of the circumcircle is x2+y2=4a2