Definition:Harmonic Conjugates/Harmonic Pencil

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


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.