In a ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n) respectively.

Then, $\frac{AB^2 +BC^2 +CA^2}{l^2 +m^2 +n^2}$

8

We know that in a ΔABC if the coordinates of the mid-points of sides BC, CA and AB are respectively $D(x_1, y_1,z_1), E(x_2, y_2, z_2)$ and $F(x_2, y_3, z_3) $, then the coordinates of its vertices are $A(-x_1+x_2+x_3, -y_1+y_2+y_3, -z_1 +z_2 +z_3),$

$B(x_1-x_2+x_3, y_1-y_2+y_3, z_1 -z_2 +z_3)$ and 

$C(x_1+x_2-x_3, y_1+y_2-y_3, z_1 +z_2 -z_3)$.

Thus, the coordinates of A, B, C are $(l, -m, n), B(l, m, -n) $ and $ C(-l, m, n)$.

$∴ \frac{AB^2 +BC^2+CA^2}{l^2+m^2+n^2}=\frac{4(m^2+n^2)+4(l^2+n^2)+4(l^2+m^2)}{l^2+m^2+n^2}= 8 $