Three holes are drilled at non-collinear points $A,$ $B,$ and $C$ in a frictionless table. A mass is placed on the interior of $\triangle ABC$ with three strings attached to it. Each string goes through a different hole and hangs downward, where it is connected to a separate hanging mass of mass $m.$ Suppose that when the mass on the table is at equilibrium, the strings leave the mass at $120^\circ$ angles to each other. Then what must be true about $\triangle ABC$?