bushindo2

The geometrical proof for the property you mentioned can be derived from two equations for a circle and one equation for the shortest distance to a line. The solution for x0 can be simplified to the quadratic equation in post #4. I suppose it is relatively straightforward to plug x0 = 2r into that quadratic equation and show that 2r is always a solution. So, do I get the coveted bonanova gold star?

Simpler solution

This is a well constructed (and well phrased) problem. Thanks bonanova for sharing. It's great to have Brainden back.