# 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

Aiming for a contradiction, suppose $AQ$ is of finite length.

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

By construction, $AP = PB$.

Then:

 $\ds \quad \dfrac {AP} {PB}$ $=$ $\ds -\dfrac {AQ} {QB}$ Definition of Harmonic Range $\ds \dfrac p p$ $=$ $\ds \dfrac q {q - 2 p}$ $\ds q - 2 p$ $=$ $\ds q$ $\ds 2 p$ $=$ $\ds 0$

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.

$\blacksquare$