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Sa se arate, pentru oricare \( z_1 \) , \( z_2 \) , \( z_3\in\mathbb{C} \), are loc inegalitatea:

\( \fbox{\ |z_1|^2+|z_2|^2+|z_3|^2\ \ge\ \mathrm{Re}(\overline{z_1}\cdot z_2)+\mathrm{Re}(\overline{z_2}\cdot z_3)+\mathrm{Re}(\overline{z_3}\cdot z_1)+\frac 16(|z_1-z_2|+|z_2-z_3|+|z_3-z_1|)^2\ } \) .

Folosim faptul ca \( |x-y|^2=|x|^2+|y|^2-2\mathrm{Re}(\overline{x}y) \). Conform acestei relatii inegalitatea se scrie echivalent :
\( |z_1-z_2|^2+|z_2-z_3|^2+|z_3-z_1|^2\ \geq \frac 13(|z_1-z_2|+|z_2-z_3|+|z_3-z_1|)^2, \)
inegalitate evidenta.