Show that three distinct complex numbers \(a,b,c\) are the vertices of an equilateral triangle if and only if \[ a^2+b^2+c^2=ab+bc+ca. \]
Solution
Set \[ u=b-a, \qquad v=c-a. \]
1. The identity\[ a^2+b^2+c^2=ab+bc+ca \]is equivalent to\[ u^2+v^2=uv. \]
Substitute \[ b=a+u, \qquad c=a+v. \] Then the left-hand side minus the right-hand side becomes \[ u^2+v^2-uv. \] Thus the two displayed equations are equivalent.
2. If the algebraic identity holds, then\[ \frac vu = e^{i\pi/3} \qquad\text{or}\qquad \frac vu = e^{-i\pi/3}. \]
Since \(a,b,c\) are distinct, in particular \[ u=b-a\neq0. \] Divide the equation from step 1 by \(u^2\) and put \[ t=\frac vu. \] Then \[ t^2-t+1=0. \] Hence \[ t = \frac{1\pm i\sqrt3}{2} = e^{\pm i\pi/3}. \]
3. If the algebraic identity holds, then \(a,b,c\) are the verticesof an equilateral triangle.
By step 2, \[ \abs v=\abs u. \] Moreover, for either choice \[ t=e^{\pm i\pi/3}, \] one has \[ \abs{t-1}=1. \] Therefore \[ \abs{v-u} = \abs u\abs{t-1} = \abs u. \] Since \[ \abs u=\abs{b-a}, \qquad \abs v=\abs{c-a}, \qquad \abs{v-u}=\abs{c-b}, \] all three side lengths are equal.
4. Conversely, if \(a,b,c\) are the vertices of an equilateraltriangle, then\[ \frac vu=e^{i\pi/3} \qquad\text{or}\qquad \frac vu=e^{-i\pi/3}. \]
Because the triangle is equilateral, \[ \abs u=\abs v=\abs{v-u}>0. \] Put \(t=v/u\). Then \[ \abs t=1 \qquad\text{and}\qquad \abs{t-1}=1. \] Write \(t=e^{i\theta}\). Squaring the second equality gives \[ 1 = \abs{e^{i\theta}-1}^2 = 2-2\cos\theta, \] so \[ \cos\theta=\frac12. \] Thus \[ t=e^{\pm i\pi/3}. \]
5. If \(a,b,c\) form an equilateral triangle, then\[ a^2+b^2+c^2=ab+bc+ca. \]
By step 4, the number \[ t=\frac vu \] satisfies \[ t^2-t+1=0. \] Multiplying by \(u^2\) yields \[ u^2+v^2=uv. \] Step 1 converts this back to the required identity.
6. Therefore the identity \(a^2+b^2+c^2=ab+bc+ca\) is necessary andsufficient for three distinct complex numbers to be the vertices ofan equilateral triangle.
Step 3 proves sufficiency and step 5 proves necessity.
7. Q.E.D.
Step 6 is the asserted equivalence.