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 $Q$.

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


 * Harmonic-range.png

We have:

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

Hence the result, by definition of harmonic conjugates.