Linejoin for fat lines?

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I draw a figure with 2 fat lines. enter image description here

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:

enter image description here

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