Two straight pipes (circular cylinders), with radii 1 and 41, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
Since ω1 and ω3 are tangent, AB=2r⋅1=2r. Since ω2 and ω3 are tangent, BC=2r⋅41=r. Since ω1 and ω2 are tangent, AC=21⋅41=1. Therefore, AC=AB+BC1=2r+r3r=1r=31r=91.
Let R be the radius of ω4.
Since ω1 and ω2 are tangent, AC=21⋅41=1. Since ω1 and ω4 are tangent, AD=2R. Since ω2 and ω4 are tangent, CD=2R⋅41=R. Therefore, AD=AC+CD2R=1+RR=1R=1. By Hint 2, r+R=91+1=910.
