If $\displaystyle \sum_{r = 1}^{4} a_{r} = 0$ and $\displaystyle \sum_{r = 1}^{4} a_{r}Z_{r} = 0$ where $\displaystyle a_{1},a_{2},a_{3},a_{4}$ are non-zero real numbers and $\displaystyle P(Z_{1}),Q(Z_{2}),R(Z_{3}), S(Z_{4})$ are concyclic points on Argand Plane, then prove that:

$\displaystyle \sum_{r = 1}^{4} a_{r}|Z_{r}|^2 = 0.48$