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