Harmonic Range/Examples/Unity Ratio

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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.


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, $AP = PB$.


\(\ds \quad \dfrac {AP} {PB}\) \(=\) \(\ds -\dfrac {AQ} {QB}\) 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.