Complex Coordinates

Using the complex plane for geometry allows us to treat points as numbers, making rotations and spiral similarities incredibly algebraic. This is often faster than Cartesian coordinates for problems involving circles and regular polygons.

Setup

The plane is the complex plane $\mathbb{C}$.

• Unit Circle: The circumcircle of $△ABC$ is usually taken as the unit circle ($|a|=|b|=|c|=1$).
• Conjugates: If $|z|=1$, then $\bar{z}=1/z$. This simplifies "reflection" logic.

Key Formulas

1. Collinearity

Points $a,b,c$ are collinear if and only if:

2. Perpendicularity

Lines $AB$ and $CD$ are perpendicular if and only if:

3. Cyclic Quadrilaterals

Points $a,b,c,d$ are concyclic if and only if their cross-ratio is real:

Properties on the Unit Circle

If $a,b,c,d$ lie on the unit circle:

• Chord Equation: The chord $AB$ connects $z$ to $a+b-ab\bar{z}$.
• Intersection of Chords: The intersection of chords $AB$ and $CD$ is:

Practice Problems