Bisection of Angle in Cartesian Plane

Theorem
Let $\theta$ be the azimuth of a point $P$ in a polar coordinate plane.

Let $QOR$ be a straight line that bisects the angle $\theta$.

Then the azimuths of $Q$ and $R$ are $\dfrac \theta 2$ and $\pi + \dfrac \theta 2$.

Proof

 * BisectedAngle.png

Let $A$ be a point on the polar axis.

By definition of bisection, $\angle AOQ = \dfrac \theta 2$.

This is the azimuth of $Q$.

Consider the conjugate angle $\complement \left({\angle AOP}\right)$ of $\angle AOP$.

By definition of conjugate angle:
 * $\complement \left({\angle AOP}\right) = -2 \pi - \theta$

where the negative sign arises from the fact that it is measured clockwise.

Then the angle $\angle AOR$ is half of $\complement \left({\angle AOP}\right)$:

The azimuth of point $R$ is the conjugate angle $\complement \left({\angle AOR}\right)$ of $\angle AOR$: