In the coordinate plane, A = (4,-1), B = (6,2), and C = (-1,2). There exists a point Q and a constant K such that for any point P,
PA2+PB2+PC2=3PQ2+k.
Find the constant K.
Thank you in advance
Let P have co-ordinates (s, t) then
PA2+PB2+PC2=(s−4)2+(t+1)2+(s−6)2+(t−2)2+(s+1)2+(t−2)2,which simplifies to3s2−18s+53+3t2−6t+9=3(s2−6s+9)+3(t2−2t+1)+32=3{(s−3)2+(t−1)2}+32.
So Q is the point with co-ordinates (3, 1) and k = 32.
Notice that the co-ordinates of Q are the average of the co-ordinates of A, B and C,
i.e. 3 = (4 + 6 - 1)/3, and 1 = (-1 + 2 + 2)/3.
Is that a coincidence ?
