Straight Line which cuts Harmonic Pencil forms Harmonic Range

Theorem

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 $\tuple {AB, PQ}$.

Let a straight line intersect $OA$, $OB$, $OP$ and $OQ$ at $A'$, $B'$, $P'$ and $Q'$ respectively.

Then $A'$, $B'$, $P'$ and $Q'$ themselves form a harmonic range $\tuple {A'B', P'Q'}$.

Proof

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Sources

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