Distance of a point in angle bicestor

Question

I believe the distance of a point in acute angle bisector is smaller than that of the obtuse angle bisector. I need to know if I know the angle between two lines is it possible to find the distance on the angle bisector which will be equal to the distance on the line?

Let us say OA = OA` = OC all of same distance. But OC will not be perpendicular to OA due to the angle. I need to know what correlation can i come up the distance CD and angle so that i can find D to make it perpendicular to OA.

Answer 1

If $\phi / 2$ is the little angle at $O$, then by one definition of cosine in a right triangle we have $$\cos\left( \frac{\phi}{2} \right) = \frac{|OA|}{|OD|}$$ so $$\cos\left( \frac{\phi}{2} \right) = \frac{|OC|}{|OD|} \quad\text{ or }\quad |OD|=\frac{|OC|}{\cos\left( \frac{\phi}{2} \right)}$$ This shows how the relative distances (from $O$) of $C$ and $D$ depend on the angle.