TITLE: If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$
QUESTION [9 upvotes]: 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?

REPLY [3 votes]: Where is the error in @Eminem 's proof?
As best as I can tell you're getting 
$$|T|=\frac{1}{R}$$ from one of three lines of thinking.  If none of these lines of thinking apply in your case, please elaborate how you derived line (2).
A.
$$(\operatorname{cis}\alpha + \operatorname{cis}\beta + \operatorname{cis}\gamma)=1$$
This is false.
or B.
$$\frac{1}{(\operatorname{cis}\alpha + \operatorname{cis}\beta + \operatorname{cis}\gamma)}=\frac{1}{\operatorname{cis}\alpha}+\frac{1}{\operatorname{cis}\beta}+\frac{1}{\operatorname{cis}\gamma}$$
This is also false.
or C
@hagen-von-eitzen correctly interpreted your thinking.  Assuming the theorem one is attempting to prove is a common mistake.
However, R does equal 1 because
$$|z1|=|z2|=|z3|=1$$
Read through 
http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf
from 
http://tutorial.math.lamar.edu/cheat_table.aspx
Particularly the Common Algebraic Errors on the last page.  Compare and contrast them with the related theorems on the first page and a half.