Angle Bisectors are Harmonic Conjugates

Theorem

Let $\LL_1$ and $\LL_2$ be straight lines which intersect at $O$.

Let $\LL_3$ and $\LL_4$ be the angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$.

Then $\LL_3$ and $\LL_4$ are harmonic conjugates with respect to $\LL_1$ and $\LL_2$.

Proof

Consider a straight line parallel to $\LL_4$ which intersects $\LL_1$, $\LL_2$ and $\LL_3$ at $L$, $M$ and $N$ respectively.

From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, $\LL_3$ is perpendicular to $\LL_4$.

Hence as $LM$ is parallel to $\LL_4$, $LM$ is perpendicular to $ON$, which is $\LL3$.

The triangle $\triangle OLM$ has:

$\angle NOL = \angle NOM$

$\angle ONL = \angle ONM$ as both are right angles

$ON$ common

So $\triangle ONL$ and $\triangle ONM$ are congruent.

So $N$ is the midpoint of $LM$.

From Harmonic Range with Unity Ratio, the points $L$, $N$, $M$ and the point at infinity form a harmonic range.

Hence from Straight Line which cuts Harmonic Pencil forms Harmonic Range, the straight lines $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ form a harmonic pencil.

That is: $\LL_3$ and $\LL_4$ are harmonic conjugates with respect to $\LL_1$ and $\LL_2$.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $22$. Equation of the bisectors of the angles between the two straight lines $a x^2 + 2 h x y + b y^2 = 0$