Condition for Cross-Ratio to be Harmonic

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Theorem

Let $A$, $B$, $C$ and $D$ be points on a straight line.

Let their cross-ratio $\set {A, B; C, D}$ be a harmonic ratio.

Then:

$\set {A, B; C, D} = \set {B, A; C, D}$


Proof

Let $\set {A, B; C, D}$ form a harmonic ratio.

We have:

\(\ds \set {A, B; C, D}\) \(=\) \(\ds -1\) Definition of Harmonic Ratio
\(\ds \leadsto \ \ \) \(\ds \dfrac {AC \cdot DB} {AD \cdot CB}\) \(=\) \(\ds -1\) Definition of Cross-Ratio of Points on Line
\(\ds \leadsto \ \ \) \(\ds \dfrac {\paren {-CA} \cdot \paren {-BD} } {\paren {-DA} \cdot \paren {-BC} }\) \(=\) \(\ds -1\) employing the fact that $AC$, $DB$ etc. are directed line segments
\(\ds \leadsto \ \ \) \(\ds \dfrac {CA \cdot BD} {DA \cdot BC}\) \(=\) \(\ds -1\) all those $-1$ factors cancel out
\(\ds \leadsto \ \ \) \(\ds \dfrac {DA \cdot BC} {CA \cdot BD}\) \(=\) \(\ds -1\) taking reciprocal of both sides
\(\ds \leadsto \ \ \) \(\ds \dfrac {BC \cdot DA} {BD \cdot CA}\) \(=\) \(\ds -1\) Real Multiplication is Commutative
\(\ds \leadsto \ \ \) \(\ds \set {B, A; C, D}\) \(=\) \(\ds -1\) Definition of Cross-Ratio of Points on Line
\(\ds \leadsto \ \ \) \(\ds \set {A, B; C, D}\) \(=\) \(\ds \set {B, A; C, D}\) as both are equal to $-1$

$\blacksquare$


Sources

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