# Equilateral triangle that has its vertices on the centers of $3$ different chords of a circle

$$A$$ is the center of the circle. The rest of the data are on the diagram. Using geogebra, it is easy to see that $$\triangle EZH$$ is equilateral, but I can't prove it. Any idea?

We will use complex numbers. Let $$\omega =e^{i\frac{\pi}{3}}\implies \omega^3=-1\implies \omega^2-\omega+1=0$$. Let $$O$$ be the center of the circle and let the circle be the unit circle. Let the points be $$a, a\omega, b, b\omega, c$$ and $$c\omega$$ in counterclockwise order as shown in the figure. Therefore $$D=\frac{a\omega+b}{2}$$ and similarly $$E$$ and $$F$$.
We wish to show that $$\triangle DEF$$ is equilateral. This is true as $$\frac{D-E}{F-E}=\frac{a\omega+b-b\omega-c}{c\omega+a-b\omega-c}=\frac{a\omega-b\omega^2+\omega^3c}{a-b\omega+\omega^2c}=\omega$$
$$\blacksquare$$