Definition:Harmonic Pencil

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


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 pencils can be found here.