I’m not using ‘Break Apart’ at all - I’m using the boolean tools. If your shape is a closed loop (which it was before you used ‘Break Apart’) you can use the boolean tools to cut it up. You will likely have to reconnect it all using ‘Auto Join’, or just re-make the original pair of circles and do it like this:
Draw a pair of circles inside each other, then draw a perfect square that starts from the center of the circle (hold the Shift key to make it a perfect square while you drag it to size):
Select both circles and group them (Ctrl + G). Select the square, then right-click and choose ‘Convert to Path’:
Click the ‘Edit Nodes’ button (just under the hexagon in the left tool bar). Hover the mouse over the the lower-right corner of the square and press D to delete that corner:
Now, select the group of two circles + the triangle, and click ‘Boolean Intersection’, like this:
It will keep the parts that overlap, leaving you with this:
