# »  Solutions to puzzles in my National Review Online Diary

## April 2009

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In my April diary I invited readers to prove Monge's theorem:

There are three circles in a plane, different sizes, none contained within either of the others. Each pair of circles has two common tangents that meet in a point. Prove that these three points lie on a straight line.

I added that:  "This is … one of those wonderful results that, with a little effort of imagination, you can 'see.' Instead of grinding your way through a page of Euclidean argument to get the proof, you can just say it in a few plain English sentences, without needing to put pencil to paper."

So how do you "see" the proof?

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Solution

I can't improve on the answer given by the engineer John Edson Sweet when shown the problem, as related in David Wells' Dictionary of Curious and Interesting Geometry:

He paused for a moment and said, "Yes, that is perfectly self-evident." Astonished, his friend asked him to explain … Prof. Sweet, in effect, replied, "Instead of three circles in a plane, imagine three balls lying on a surface plate. Instead of drawing tangents, imagine a cone wrapped around each pair of balls. The apexes of the three cones will then lie on the surface plate. On top of the balls lay another surface plate. It will rest on the three balls and will be necessarily tangent to each of the three cones, and will contain the apexes of the three cones. Thus the apexes of the three cones will lie in both of the two plates, which is of course a straight line."