If $\sum _{\text{i=1}}^{\text{4}}\left({\text{x}}_{\text{i}}^{\text{2}}{\text{+y}}_{\text{i}}^{\text{2}}\right)\le {\text{2x}}_{\text{1}}{\text{x}}_{\text{3}}{\text{+2x}}_{\text{2}}{\text{x}}_{\text{4}}{\text{+2y}}_{\text{2}}{\text{y}}_{\text{3}}{\text{+2y}}_{\text{1}}{\text{y}}_{\text{4}}$  then points  $\left({\text{x}}_{\text{1}}{\text{,\hspace{0.17em}y}}_{\text{1}}\right)\text{,\hspace{0.17em}}\left({\text{x}}_{\text{2}}{\text{,\hspace{0.17em}y}}_{\text{2}}\right)\text{,}\left({\text{x}}_{\text{3}}{\text{,\hspace{0.17em}y}}_{\text{3}}\right)\text{,}\left({\text{x}}_{\text{4}}{\text{,\hspace{0.17em}y}}_{\text{4}}\right)$ are

On solving we have
${\left({\text{x}}_{\text{1}}{\text{-x}}_{\text{3}}\right)}^{\text{2}}\text{+}{\left({\text{x}}_{\text{2}}{\text{-x}}_{\text{4}}\right)}^{\text{2}}\text{+}{\left({\text{y}}_{\text{2}}{\text{-y}}_{\text{3}}\right)}^{\text{2}}\text{+}{\left({\text{y}}_{\text{1}}{\text{-y}}_{\text{4}}\right)}^{\text{2}}\le 0$
$\Rightarrow {\text{x}}_{\text{1}}{\text{= x}}_{\text{3}}{\text{,\hspace{0.17em}x}}_{\text{2}}{\text{= x}}_{\text{4}}{\text{,\hspace{0.17em}y}}_{\text{2}}{\text{= y}}_{\text{3}}{\text{\hspace{0.17em}and\hspace{0.17em}y}}_{\text{1}}{\text{= y}}_{\text{4}}$
$\Rightarrow \frac{{\text{x}}_{\text{1}}{\text{+ x}}_{\text{2}}}{\text{2}}\text{=}\frac{{\text{x}}_{\text{3}}{\text{+ x}}_{\text{4}}}{\text{2}}\text{\hspace{0.17em}and\hspace{0.17em}}\frac{{\text{y}}_{\text{1}}{\text{+ y}}_{\text{2}}}{\text{2}}\text{=}\frac{{\text{y}}_{\text{4}}{\text{+ y}}_{\text{3}}}{\text{2}}$
Hence AB and CD bisect each other
Also AC| |BD and AD| |BC Which are both parallel to co-ordinate axis.