Monge's Circles

Take any three circles. Each pair has two outer tangent lines, and those tangents cross at a point. Monge's theorem says the three crossing points always sit on one straight line. Drag the circles and watch the line hold.

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Why is it always true?

Imagine each circle is the equator of a ball resting on this page. The two outer tangents of a pair of circles are the outline of a cone that wraps snugly around both balls, and the point where the tangents cross is the cone's tip.

Now lay a flat sheet on top of all three balls. It touches each ball once and contains every cone's tip, because a cone that hugs two balls fits under the sheet too. The three tips lie on the sheet and also on the page, so they lie where the two planes meet: a single straight line.

The crossing points are also called external homothetic centers. Switch on the internal centers to see d'Alembert's companion result: one external center and the other two internal centers are collinear as well.