# Definition:Harmonic Conjugates/Harmonic Pencil

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*This page is about Harmonic Conjugates. For other uses, see Harmonic.*

## Definition

Let $AB$ and $PQ$ be line segments on a straight line such that $\tuple {AB, PQ}$ is a **harmonic range**.

Let $O$ be a point which is not on the straight line $AB$.

Let $\map O {AB, PQ}$ be the **harmonic pencil** formed from $O$ and $\tuple {AB, PQ}$.

The rays $OP$ and $OQ$ are said to be **harmonic conjugates with respect to $OA$ and $OB$**.

## Also known as

The two pairs of rays $\set {OA, OB}$ and $\set {OP, OQ}$ can also be said to be **apolar**.

## Also see

- Results about
**Harmonic Ranges**can be found**here**.

## Sources

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $19$. Harmonic ranges and pencils