The Indiana University Bloomington mathematics department has a rendition of the Seven Circle Theorem on their mugs. While visiting, I came up with a synthetic geometric proof. I'll give some hints in the replies. https://en.wikipedia.org/wiki/Seven_circles_theorem …https://www.youtube.com/watch?v=kPJURKUg13c …
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The first hint: three lines connecting the tangency points of opposite circles can be thought of as geodesics in the Klein model of the hyperbolic plane. https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model …
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Whereas the 6 tangent circles may be thought of as horocycles in the Poincaré disc model of the hyperbolic plane. https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model … https://en.wikipedia.org/wiki/Horocycle
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Three geodesic lines will intersect in the Klein model if and only if three geodesic lines intersect in the Poincaré disc model. Hence we may replace the three lines with three circles orthogonal to the boundary of the outer circle, and try to show that these intersect.
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Now the configuration is invariant under linear fractional transformations preserving the Poincaré disc, i.e. isometries of the hyperbolic plane. So we may perform such a transformation sending two of the geodesics to intersect at the origin, in which case they are now straight.
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For the cycle of 6 horocycles, we may shrink every other one by hyperbolic distance r, and increase the other three by r, while maintaining the cycle of tangencies. Note that opposite horocycles will maintain their distance under this transformation.
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Recall that two of the three geodesics are straight lines meeting at the origin. Resize the horocycles so that one opposite pair has the same size. By symmetry (a π rotation in the origin), the other pair will as well.
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Now one also sees by symmetry that third pair of horocycles has equal size and are related by rotation through π in the origin. Hence the Poincaré geodesic connecting the tangency points is a line through the origin, confirming the theorem.
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This is indeed a lovely argument. Thank you very much for sharing it here. And thank you, too, for making the theorem itself better known in this way; this was my introduction to it.
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