Let P be (x,y). Then PA^2 + PB^2 + PC^2 = = (x-4)^2 + (y+1)^2 + (x+3)^2 + (y-1)^2 + (x-5)^2 + (y+3)^2 = = 3x^2 + 3y^2 - 12x + 6y + 61 = =3(x^2 + y^2 - 4x + 2y) + 61 = = 3((x-2)^2 + (y+1)^2) + 46. This shows that if Q = (2,-1), then PQ^2 = (x-2)^2 + (y+1)^2. And so PA^2 + PB^2 + PC^2 = 3PQ^2 + 46. ANSWER. k = 46.