Leaving the plane

Living on the Earth surface people thought for a long time that it was flat. We had to build some theo­ries to guess that the Earth looks more like a ball. And only in the second part of the XX century we could look at our planet from the space and certify it visu­ally.

The same happens in math­e­matics: consid­ering the ambient space we can often learn more about some object.

Consider three arbi­trary circles and draw common tangents to every pair. What can we say about the single line. Well, a picture is not a proof, it's just a source for our conjec­ture. Let's try to prove it.

Both the problem and the image consider plane objects. But what if we look at them from outside, from the ambient three-dimen­sional space.

Consider three spheres whose equa­tors are the circles. The cones enveloping the pairs of spheres have the common tangents as gener­a­trices. The points we expect to lie on the same line are the apices of the cones.

Put a plane on the cones. The upper gener­a­trices inter­sect pair­wise and define this plane unam­bigu­ously. The points we a inter­ested in, the cones apices, lie both on this plane and the «equa­to­rial» one. But this two nonpar­allel planes have inter­sect in a line! So, as we conjec­tured, the three points of inter­sec­tion of common tangents to three arbi­trary circles lie on a single line.

Nowa­days this theorem we've just proved is named after a french math­e­mati­cian Gasr­pard Monge.

Other etudes in “Varios interesting subjects”