$z_1,z_2,z_3 \in \mathbb{C}$ Prove that $z_1^2+z_2^2+z_3^2=z_1z_2+z_1z_3+z_2z_3$ is true iff $z_1,z_2,z_3$ are points of a equilateral triangle.
130109
Equilateral Triangle from three complex points
82777
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
142835
Proving :$\arctan(1)+\arctan(2)+ \arctan(3)=\pi$
140977
Why $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
39779
If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
126385
How to prove $a^3 + b^3 +c^3 - 3abc =1 $, If a,b,c are infinite series as given in the description box below?
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
97057
Show $a^2+b^2+c^2=1$ $\implies$ $ \dfrac {b^2c^2}{1+a^2} +\dfrac {c^2a^2}{1+b^2}+\dfrac {a^2b^2}{1+c^2} \le \dfrac 14 $
144335
Sum of three numbers and the sum of their squares: $x^2+y^2+z^2 \geq \frac{1}{3}$ for $x+y+z=1$
59239
For non-zero real $x$, $y$, $z$ and complex $a$, $b$, $c$ with $|a|=|b|=|c|$, if $x+y+z=0=ax+by+cz$, then $a=b=c$
102320
Is my solution correct for solving $\cos(z)=2$, $z\in \mathbb{C}$
123133
If, in a triangle, $\cos(A) + \cos(B) + 2\cos(C) = 2$ prove that the sides of the triangle are in AP
64096
Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
139065
Proving $\csc\frac\alpha2+\csc\frac\beta2+\csc\frac\gamma2 \ge 6$, where $\alpha$, $\beta$, $\gamma$ are the angles of a triangle
62297
What can be said about complex numbers $z_1, z_2, z_3$ if $\frac{z_1 - z_3}{z_2 - z_3}$ is real?
28218
Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$