Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1=r$ which $r$ is a real number.
140977
Why $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$
139168
Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number
16774
If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
68567
Prove that every complex number with modulus 1 and is not -1, has this property
86197
Let $z_1 = x_1 +iy_1 $ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $ = x_1x_2+y_1y_2$
83359
Complex number identity by trigonometry
56341
Solving complex numbers equation $z^3 = \overline{z} $
90549
If $z^2 = z_1^2+z_2^2+z_3^2$ then $|\Re(z)| \le|\Re(z_1)|+|\Re(z_2)|+|\Re(z_3)|$
20305
Complex roots of $z^3 + \bar{z} = 0$
82085
Without solving the equation, show that all nine roots of $(𝑧 − 1)^{10} = 𝑧^{10}$ lie on the line $Re (𝑧) = \frac{1}{2}$
147986
Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number
116223
Show that the angles satisfy $x+y=z$
101491
If $\frac{a}{|z_2-z_3|}=\frac{b}{|z_3-z_1|}=\frac{c}{|z_1-z_2|}$, then why is $\frac{a^2}{z_2-z_3}+\frac{b^2}{z_3-z_1}+\frac{c^2}{z_1-z_2}=0$?
36980
Prove the third isomorphism theorem
71540
Real and Imaginary Parts of tan(z)
156782
Confused about complex numbers
148178
What points in the complex plane satisfy $|z|=arg(z)$.