+1791 Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
\[PA^2 + PB^2 + PC^2 = 3PQ^2 + k.\]
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant $k$.
+128475 Let P= (x,y)
PA^2 + PB^2 + PC^2 =
(x - 7)^2 + (y + 11)^2 + ( x-10)^2 + (y -13)^2 + ( x-18)^2 + (y + 22)^2 =
3x^2 - 70x + 3y^2 + 40y + 1247
3 [ x^2 - (70/3x + y^2 +(40/3)y + 1247/3) ] complete the square on x,y
3 [ (x^2 - (70/3)x + 1225/9 + y^2 + (40/3)y + 400/9 + 1247/3 - 1225/9 - 400/9)
3 [ (x - 35/3)^2 + (y + 20/3)^2 ] + 3 [ 1247/3 -1225/9 - 400/9 ]
3 [ (x - 35/3)^2 + (y + 20/3)^2 ] + 2216/3
Q = ( 35/3 , -20/3)
k = 2216 / 3
