How to find an intersection between a normal that goes through a vertex and 2 polylines that are on the sides of the polyline containing that vertex?

Question

This is a little hard to explain, so let's start with a picture:

What I need to do is to find x,y coordinates of the points R1 and R2.

Coordinates of every other point are known.

I am trying to implement this in C#, so the optimal solution would be a function that takes x,y coordinates of known points and calculates x,y coordinates of the unknown points. I tried to combine several methods to solve this, but I failed.

note: there can be a case when the angle α/2 is 90°

Answer 1

Write the implicit equations of the lines $AB$ and $BC$, in the form $ax+by+c=0$, where the coefficients are normalized so that $a^2+b^2=1$. Then add the two equations to get the equation of the bissectrix.

You will find the intersecting segments by plugging the endpoint coordinates in that equation and observing a change of sign.