no ones any suggestions?

I'm having some ideas, why not share them...

1. I first though about using two transparent colors and thus getting a third mix-color to distinguish shapes. But from your description you want to know points, rather than just see/get the (in this case) three shapes.

2. So my second thought was, what about a 'cumulative line-matrix' or in other words a 2-dimensional array with dimensions of your window => int[width][height]

Example

1. 0000000000000
2. 0011111100000
3. 0010000100000
4. 0010111211100
5. 0010100100100
6. 0011211100100
7. 0000100000100
8. 0000111111100
9. 0000000000000

In such an array the number would correspond with the number of lines in that spot:
0 = zero lines
1 = one line (regular)
2 = two lines (intersection!)
3+ = three or more lines (multiple intersections)

I don't know what directly the practical application of this would be, but it seems like an interesting way to determine all intersections for the whole image in one sweep, instead of brute-forcing each point on every rectangle with each other.

3. Finally, isn't it pretty easy to just calculate instead of brute-force the intersection? For example point 9 has the horizontal (X) value of either point 2 or 4 with the vertical (Y) value of either 5 or 6 right? Or do you not have the original coordinates either?

So I guess there are different ways to get the shapes and the intersections. Or maybe do a mixture of techniques... Getting the desired output like that is another step of course.