Let the distance of the centroid from each vertex is r and it can be written as,
r=l√3
The gravitational potential at centroid due to mass m at A is given as,
VA=−Gmr
=−√3Gml
The gravitational potentials at B and C are given as,
VB=−√3Gml
VC=−√3Gml
The total potential due to all the three masses is given as,
V=VA+VB+VC
V=−√3Gml−√3Gml−√3Gml
V=−3√3Gml
Thus, the gravitational potential at the centroid of the triangle is −3√3Gml.