# Definition:Harmonic Pencil

This page is about Harmonic Pencil. For other uses, see Harmonic.

## Definition

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

Let $P$ and $Q$ lie on $AB$ such that $\tuple {AB, PQ}$ is a harmonic range.

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

Then the pencil $\map O {AB, PQ}$ formed by joining $O$ to the four points $A$, $B$, $P$ and $Q$ is said to be a harmonic pencil.

## Also see

• Results about Harmonic Ranges can be found here.