Harmonic Range/Examples/Unity Ratio

Examples of Harmonic Ranges
Let $A$, $B$, $P$ and $Q$ be points on a straight line.

Let $\tuple {AB, PQ}$ be a harmonic range such that $P$ is the midpoint of $AB$.

Then $Q$ is the point at infinity.

Proof
$AQ$ is of finite length.

Let $p := AP$ and $q := AQ$ be the (undirected) lengths of $AP$ and $AQ$ respectively.

By construction, $AP = PB$.

Then:

That is, the length of $AP$ is zero.

which contradicts the definition of $AP$.

Hence $AQ$ can not be of finite length.

Hence the result.