Hi Ed,

I was suspicious about your solution. and yes I think PAQUICINNO's method works better maybe! ( yet it needs to be proved mathematically)

And I checked something when there is a line tangent to a circle and when you zoom in on atomic scale you see that there is a tiny gap between tangent line and the circle - which I believe it is normal to see such thing at such microscopic zooms - but there are no several crossings!

And feel free to check what I said in the attached file.