How to draw a combination of circles.

Hi

Do you have Illustrator? If so, it's super simple:

1. Select your circles with the Selection tool
   
   
   
2. Press Shift+M to get the Shape Builder tool and hover over what are now five regions
   
3. Option+Click the regions you don't want
   
4. You will be left with the region(s) you do want
   
5. You can keep it in Illustrator or select it with the Selection tool and copy and paste it to Photoshop.
   
6. I chose Path. It will need to be stroked on a new layer.
   
   

It looks like you said you want the inner lines. Is this what you are looking for?

~ Jane

There may be a better way to do this but...

If it were me, I would keep all three circles on separate layers.  I would then erase the parts of the two outside circles that you do not want using the eraser tool, and then merge all three layers.

Hope that helps. 

Hi

One way

Draw three circle shapes the left and right with no fill and the bottom with white fill. Clip the upper shape layers to the bottom shape layer

Dave