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.
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.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $19$. Harmonic ranges and pencils