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

Question

$A$ 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?

Answer 1

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 $$\frac{D-E}{F-E}=\frac{a\omega+b-b\omega-c}{c\omega+a-b\omega-c}=\frac{a\omega-b\omega^2+\omega^3c}{a-b\omega+\omega^2c}=\omega$$

$\blacksquare$