Harmonic Conjugacy is Symmetric

Theorem

Let $AB$ and $PQ$ be line segments on a straight line.

Let $P$ and $Q$ be harmonic conjugates with respect to $A$ and $B$.

Then $A$ and $B$ are harmonic conjugates with respect to $P$ and $Q$ .

By definition of harmonic conjugates, $\tuple {AB, PQ}$ is a harmonic range.

We have:

$\ds \dfrac {AP} {PB}$

$=$

$\ds -\dfrac {AQ} {QB}$

Definition of Harmonic Range

$\ds \leadsto \ \ $

$\ds -\dfrac {PA} {PB}$

$=$

$\ds -\paren {-\dfrac {AQ} {BQ} }$

reversing the direction of $AP$ and $QB$

$\ds $

$=$

$\ds \dfrac {AQ} {BQ}$

$\ds \leadsto \ \ $

$\ds -PA$

$=$

$\ds \dfrac {AQ \times PB} {BQ}$

multiplying by $PB$

$\ds \leadsto \ \ $

$\ds -\dfrac {PA} {AQ}$

$=$

$\ds \dfrac {PB} {BQ}$

dividing by $BQ$

Hence, by definition, $\tuple {PQ, AB}$ is a harmonic range.

Hence the result, by definition of harmonic conjugates.

$\blacksquare$

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