Modulli and complex conjugates

Problem:

Let $z_{1}$ , $z_{2}$ , $z_{3}$ be complex numbers such that

$|z_{1}|=|z_{2}|=|z_{3}|=r>0$

and $z_{1}+z_{2}+z_{3} \neq 0$ . Prove that

$|\frac{z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}}{z_{1}+z_{2}+z_{3}}|=r$

Proof:

Observe that $z_{1}.\overline{z_{1}}=z_{2}.\overline{z_{2}}=z_{3}.\overline{z_{3}}=r^{2}$

Then,

$|\frac{z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}}{z_{1}+z_{2}+z_{3}}|^{2}=\frac{z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}}{z_{1}+z_{2}+z_{3}}.\frac{\overline{z_{1}z_{2}}+\overline{z_{2}z_{3}}+\overline{z_{3}z_{1}}}{\overline{z_{1}}+\overline{z_{2}}+\overline{z_{3}}}$

This equals

$\frac{z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}}{z_{1}+z_{2}+z_{3}}.\frac{\frac{r^{2}.r^{2}}{z_{1}.z_{2}}+\frac{r^{2}.r^{2}}{z_{2}.z_{3}}+\frac{r^{2}.r^{2}}{z_{3}.z_{1}}}{\frac{r^{2}}{z_{1}}+\frac{r^{2}}{z_{2}}+\frac{r^{2}}{z_{3}}}=r^{2}$

as desired.

QED.

More later,

Nalin Pithwa

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