Without loss of generality, assume we have a unit square. If two non-parallel lines intersect within this square, there will be four points on the boundary of the square. The lines and the boundary of the square create four regions.
¿Is there a (nice) formula to find the areas of the 4 distinct regions of the square given the boundary intersection points?
Follow-up query: I am looking at this for a needle-craft art project, but as a former mathematics instructor, I’d be curious to know of any other applications for a problem such as this.
If you have the 4 boundary points, and your square has corners $(0,0), (0,1), (1,0), (1,1)$, you can construct the two equations of the lines for your non-parallel lines. You'll also need the intersection of the two lines, call it $(a,b)$.
Integrating the line that goes from the bottom left to the middle from the point on the boundary of the square to $(a,b)$ and then adding the integral of the line that goes from the middle to the bottom right from $(a,b)$ to the point on the boundary will give the area of the bottom of the four sections.
You can just rotate your square by 90 degrees each time and redo this process to find the other three areas.