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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 $A = (2,4),$ $B = (-3,1),$ and $C = (1,7)$, then find the constant $k$.

 
 Apr 14, 2024

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