Answer to Question #223143 in Math for mungaster

Question

Consider 3 distinct points M₁, M₂, and M₃ with respective affixes z₁, z₂, and z₃. Show that (Triangle M₁, M₂, and M₃ is equilateral) ↔ ( z₁² + z₂² +z₃² - z₁z₂ - z₂z₃ - z₁z₃).

Answer 1

The problem is similar to this problem:

Given three complex numbers $z_{1}, z_{2}, z_{3}$ prove that the points $z_{1}, z_{2}, z_{3}$ are vertices of an equilateral triangle in $\mathbb{C}$ , if $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{1} z_{3}+z_{2} z_{3}$  .

Three points $z_{1}, z_{2}, z_{3}$ are vertices of an equilateral triangle

 

$\begin{aligned} &\Longleftrightarrow \frac{z_{3}-z_{1}}{z_{2}-z_{1}}=\cos \left(\pm 60^{\circ}\right)+i \sin \left(\pm 60^{\circ}\right)=\frac{1 \pm \sqrt{3} i}{2} \\ &\Longleftrightarrow \quad 2 z_{3}-z_{1}-z_{2}=\pm \sqrt{3} i\left(z_{2}-z_{1}\right) \\ &\Longleftrightarrow\left(2 z_{3}-z_{1}-z_{2}\right)^{2}=\left(\pm \sqrt{3} i\left(z_{2}-z_{1}\right)\right)^{2} \\ &\Longleftrightarrow \quad z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_3+z_{3} z_{1} \end{aligned}$