This is currently 3 circles and 2 lines however I would like to split it into lots of small paths on wherever lines intersect as shown in the second image.
How can I do this either with Inkscape or another tool?
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You should be able to do this with the Cut Path operation.
Firstly, you have to make sure your lines extend just beyond your circles or Inkscape can't cut the path properly.
Next, select all three circles by holding shift then clicking each individually. Once all three circles are selected, press Ctrl + K (or Path menu / Combine) The treats the circles as one object.
Now do the same thing with your lines. Hold shift and click both lines, then press Ctrl + K. (Note, it's possible to use one line at a time, but combining them is quicker. However if it suits your purpose to cut with each line individually, you can certainly do that.)
Make sure that your lines are on top of your circles, or you'll cut the wrong pieces! Click the lines once and press the 'Raise to top button'.
Finally, click your lines once, if they aren't already selected, then click your circles. With both objects selected, on the Path menu, click Cut Path (or Ctrl + Alt + /)
This breaks the circles into pieces, but your lines will disappear. If you need the lines to remain, duplicate them before performing the Cut Path operation (Ctrl + D). Also, if you need the lines to be separate objects again at the end, press Ctrl + Shift + K, or click Break Apart on the Path menu (this is the opposite of the Combine command we did earlier).
I hope that gets you what you want. I've included an image below to show the steps.
Unfortunately Inkscape hasn't a direct single command for "cut everything in every crossing".
A workaround: Cut the lines by circles and circles by lines using Path > Cut Path
I'm not fully aware when the cutter should vanish and when there should be left some remnants. The recoloring is for identifying the unwanted remnants.
ADDENDUM: Just noticed that the lines can be a little short. To be sure they are not too short it's good to drag them a little longer. The exessive parts are easy to delete after the cutting.