I draw a figure with 2 fat lines.

I need to draw a join between these lines correctly. Long red lines are in a middle of each fat line.
What I know:
- coordinates of white points.
- the angle between these 2 fat lines could be different, from 0 to Pi (not always Pi/2), but the thickness of these lines is always the same.
- Red lines are not drawn.
I need to find coordinates of a black point. How to solve this mathematical problem?
Edited
Solution with vectors is incorrect or I haven't understood it:

It shows what I currently have with vector addition. Blue lines means the solution I want to achieve and their intersection is the same as black point from a previous picture