Show that the point P of the triangle of Problem 31 that minimizes |AP| + |BP| +

Question:

Show that the point P of the triangle of Problem 31 that minimizes |AP| + |BP| + |CP| is the point where the three angles at P are equal. Let A', B', and C' be the points where the weights are attached. The center of gravity is then located 1/3(|AA'| |BB'| + + |CC'|) units below the plane of the triangle. The system is in equilibrium when the center of gravity of the three weights is lowest.