Bisection of Angle in Cartesian Plane

Theorem

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

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

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

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

This is the angular coordinate of $Q$.

$\Box$

Consider the conjugate angle $\map \complement {\angle AOP}$ of $\angle AOP$.

By definition of conjugate angle:

$\map \complement {\angle AOP} = -2 \pi - \theta$

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

Then the angle $\angle AOR$ is half of $\map \complement {\angle AOP}$:

$\ds \angle AOR$

$=$

$\ds -\paren {\frac {2 \pi - \theta} 2}$

$\ds $

$=$

$\ds -\pi + \frac \theta 2$

The angular coordinate of point $R$ is the conjugate angle $\map \complement {\angle AOR}$ of $\angle AOR$:

$\ds \map \complement {\angle AOR}$

$=$

$\ds 2 \pi - \paren {-\pi + \frac \theta 2}$

$\ds $

$=$

$\ds 2 \pi - \pi + \frac \theta 2$

$\ds $

$=$

$\ds \pi + \frac \theta 2$

$\blacksquare$