# Category:Harmonic Ranges

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This category contains results about **Harmonic Ranges**.

Definitions specific to this category can be found in Definitions/Harmonic Ranges.

Let $A$ and $B$ be points on a straight line.

Let $P$ and $Q$ lie on $AB$ such that $P$ is on the line segment $AB$ while $Q$ is outside the line segment $AB$.

Let $P$ and $Q$ be positioned such that the **cross-ratio** $\set {A, B; P, Q}$ forms a harmonic ratio:

- $\dfrac {AP} {PB} = -\dfrac {AQ} {QB}$

Then $\tuple {AB, PQ}$ are said to be a **harmonic range**.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### H

- Harmonic Conjugates (empty)
- Harmonic Pencils (empty)

## Pages in category "Harmonic Ranges"

The following 9 pages are in this category, out of 9 total.