# Definition:Harmonic Conjugates

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## Definition

### Harmonic Range

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

Then $P$ and $Q$ are said to be **harmonic conjugates** with respect to $A$ and $B$.

### Harmonic Pencil

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

Two pairs of:

which are **harmonic conjugates** are also known as a **conjugate pair**.

**Harmonic conjugates** can also be said to be **apolar**.

## Also see

- Results about
**harmonic conjugates**can be found**here**.