Bisectors of Angles between Two Straight Lines/Homogeneous Quadratic Equation Form

Theorem
Consider the homogeneous quadratic equation:


 * $(1): \quad a x^2 + 2 h x y + b y^2 = 0$

representing two straight lines through the origin.

Then the homogeneous quadratic equation which represents the angle bisectors of the angles formed at their point of intersection is given by:


 * $h x^2 - \paren {a - b} x y - h y^2 = 0$

Proof
From Angle Bisectors are Harmonic Conjugates, the two angle bisectors are harmonic conjugates of the straight lines represented by $(1)$.

From Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines, these angle bisectors can be described by the homogeneous quadratic equation:
 * $x^2 + 2 \lambda x y - y^2 = 0$

From Condition for Pairs of Lines through Origin to be Harmonic Conjugates: Homogeneous Quadratic Equation Form:


 * $-a + b - 2 \lambda h = 0$

Hence:
 * $\lambda = -\dfrac {a + b} {2 h}$

The result follows.