# Harmonic Conjugacy is Symmetric

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## 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 $Q$ .

## Proof

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.

$\blacksquare$