three point charges each ‘q’ is places at the three vertices of an equilatral triange.find the magnitude and sign of the charge to be placed at the centroid of the triange so that the vertices are in equilibrium.

If we put positive charge at the centre, due to same polarity interaction, they will repel each other.
If negative, then due to attraction by three charge it would sustained its position.
Now, for the system to be in equilibrium, calculate the potential energy and double negative differential of the P.E. w.r.t. position must be zero.
Proceed in this way and calculate the energy.