Overview

Multiplication by $e^{i\theta}$ rotates points, and addition translates them. This gives a compact way to solve plane geometry problems.

Core Skills

Use Rotation Multiplication

If a rotation by $\theta$ is present, multiply by $e^{i\theta}$ and compare ratios of complex differences.

Translate First

Shift the origin to simplify: replace $z$ with $z-z_0$ so rotations are about the origin.

Compare Distances

Use $|z_1-z_2|$ to encode distance constraints in the plane.

Key Ideas

• Rotation by $\theta$: $z \mapsto ze^{i\theta}$.
• Translation by $w$: $z \mapsto z + w$.
• Distance: $|z_1 - z_2|$.

Worked Example

How many nonzero $z$ make $0, z, z^3$ the vertices of an equilateral triangle?

Equilateral means $(z-0)/(z^3-0) = z^{-2}$ is a cube root of unity $\omega$ or $\omega^2$. So $z^2$ equals $\omega$ or $\omega^2$, giving four solutions on the unit circle. The answer is $4$.

More Examples

Example 1: Midpoint

If $A$ and $B$ correspond to $z_1$ and $z_2$, the midpoint is $(z_1+z_2)/2$.

Example 2: Rotation About a Point

Rotate point $z$ about $w$ by $\theta$.

The image is $w + (z-w)e^{i\theta}$.

Example 3: Perpendicularity

If $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{CD}$, then $(z_B-z_A)/(z_D-z_C)$ is purely imaginary.

Strategy Checklist

• Translate the figure so a rotation center is at $0$.
• Use ratios of differences to encode angles.
• Apply modulus for distances.

Common Pitfalls

• Forgetting to divide out the common vertex before applying rotation criteria.
• Mixing up $\omega$ with $\omega^2$ in equilateral conditions.
• Failing to translate before applying rotation formulas.

Practice Problems