Rotation About an Arbitrary Point

Rotating z by angle α about the point z₀: shift z₀ to origin, rotate, shift back. Rearranged: z' = z₀ + (z−z₀)e^(iα). This is the general form of the Coni method and is used to find the third vertex of an equilateral triangle, rotate a line, or construct geometric figures.

Rotation about the origin is simple: multiply by $e^{i\alpha}$. Rotation about any other point uses one extra step on each side.

The Three-Step Method

To rotate point $z$ by angle $\alpha$ about the centre $z_0$:

Step 1 — Translate $z_0$ to the origin.

Replace $z$ with $z - z_0$. In this shifted frame, the centre of rotation is now at $0$.

Step 2 — Rotate about the origin.

Multiply by $e^{i\alpha}$:

$$(z - z_0) \xrightarrow{\times e^{i\alpha}} (z - z_0)e^{i\alpha}$$

Step 3 — Translate back.

Add $z_0$ to restore the original coordinate system:

$$z' = z_0 + (z - z_0)e^{i\alpha}$$

The Formula

$$\boxed{z' - z_0 = (z - z_0)\,e^{i\alpha}}$$

Written this way, the structure is clear: the displacement from the centre transforms by multiplication with $e^{i\alpha}$.

Application — Third Vertex of an Equilateral Triangle

Given two vertices $z_1$ and $z_2$, find the third vertex $z_3$ of an equilateral triangle.

The third vertex is obtained by rotating $z_2$ about $z_1$ (or $z_1$ about $z_2$) by $\pm 60°$:

$$z_3 - z_1 = (z_2 - z_1)\,e^{\pm i\pi/3}$$

Since $e^{i\pi/3} = \cos 60° + i\sin 60° = \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i$, the two possible third vertices are:

$$z_3 = z_1 + (z_2 - z_1)\left(\frac{1 \pm i\sqrt{3}}{2}\right)$$

The $+$ gives the vertex on the left of $z_1 z_2$; the $-$ gives the vertex on the right.

Connection to the Coni Method

The Coni method formula $\dfrac{z_3 - z_1}{z_2 - z_1} = \dfrac{|z_3-z_1|}{|z_2-z_1|}e^{i\alpha}$ is the same statement in ratio form. When $|z_3 - z_1| = |z_2 - z_1|$, it reduces exactly to the rotation formula above with the ratio equal to $e^{i\alpha}$.