Our teacher gave us a hard question (according to her, it is pretty hard for our level):
Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$.
Now, the class tried for like 40 minutes to prove that, and then the teacher came up with some really complicated proof.
I sat quietly and came up with this proof:
$$|z_1+z_2+z_3| = |R(\operatorname{cis}\alpha + \operatorname{cis}\beta + \operatorname{cis}\gamma)| = I$$
Thus
$$|I| = R\tag{1}$$
Also,
$$\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=\left|\frac{1}{R}\left(\frac{1}{\operatorname{cis}\alpha}+\frac{1}{\operatorname{cis}\beta}+\frac{1}{\operatorname{cis}\gamma}\right)\right| = T$$
Thus
$$|T| = \frac{1}{R} \tag{2}$$
It's easy to see that from $(1)$ and $(2)$ we get:
$$R=\frac{1}{R}$$
Thus
$$\frac{1}{1} = 1$$
Which finish the proof.
My teacher said that there is a mistake in my proof, but found none – she said it could not be that easy.
Is there an error in my proof? Or is it valid?
Best Answer
The following is a fairly easy proof.
Since $z_1,z_2,z_3$ are on the unit circle, we have:$$\frac{1}{z_i}=\overline{z_i}\qquad i=1,2,3.$$ Hence,$$\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=\left|\overline{z_1}+\overline{z_2}+\overline{z_3}\right|=|\overline{z_1+z_2+z_3}|=|z_1+z_2+z_3|.$$