Condition for Pairs of Lines through Origin to be Harmonic Conjugates

Theorem
Consider $4$ lines $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ through the origin $O$ whose equations embedded in the Cartesian plane are as follows:

Then the pairs of lines $\set {\LL_1, \LL_2}$ and $\set {\LL_3, \LL_4}$ are harmonic conjugates :


 * $\dfrac {\lambda - \lambda'} {\lambda - \mu'} / \dfrac {\mu - \lambda'} {\mu - \mu'} = -1$

Proof
Let $\set {\LL_1, \LL_2}$ and $\set {\LL_3, \LL_4}$ be harmonic conjugates as asserted.

A straight line in the plane which does not pass through $O$ will either:
 * intersect all four of $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$

or:
 * be parallel to one such straight line and intersect the other three.

Hence, let a straight line be drawn parallel to $\LL_1$ so as to intersect each of $\LL_2$, $\LL_3$, $\LL_4$ at $L'$, $M$ and $M'$.

It would cut $\LL_1$ at $L$, but $L$ is the point at infinity of $\LL_1$.



From Straight Line which cuts Harmonic Pencil forms Harmonic Range, the points of intersection define a harmonic range.

We have that $L'M'$ is parallel to $\LL_1$.

From Harmonic Range with Unity Ratio, the harmonic range defined by $L'M'$ is such that $M$ is the midpoint of $L'M'$.

$L'M'$ can be expressed in slope-intercept form as $y = \lambda x + c$ for some $c \in \R_{\ne 0}$.

Hence the abscissae of the points $L'$, $m$ and $M'$ are respectively:
 * $\dfrac c {\lambda' - \lambda}$, $\dfrac c {\mu - \lambda}$, $\dfrac c {\mu' - \lambda}$

Hence the condition that the pairs of lines $\set {\LL_1, \LL_2}$ and $\set {\LL_3, \LL_4}$ are harmonic conjugates is:


 * $\dfrac 2 {\mu - \lambda} = \dfrac 1 {\lambda' - \lambda} + \dfrac 1 {\mu' - \lambda}$

which is more usually expressed as:


 * $\dfrac {\lambda - \lambda'} {\lambda - \mu'} / \dfrac {\mu - \lambda'} {\mu - \mu'} = -1$

Also presented as
This result can also be presented as:


 * $\dfrac {\paren {\lambda - \lambda'} \paren {\mu - \mu'} } {\paren {\lambda - \mu'} \paren {\mu - \lambda'} } = -1$