Let D be a triangle in a hyperbolic plane. Prove that there exist points A,B,C on three different sides of D such that diameter of the set {A,B,C} is bounded by a constant which is independent from the choice of D
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Hint: consider the largest possible triangle. How would you choose the points to minimize diameter? How can this help for other triangles? – MvG Nov 01 '17 at 22:54
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@MvG: I understand that the largest triangle is important from the point of view of this question. But how would you choose the three points? – zoli Nov 02 '17 at 02:24
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I suggest you draw an ideal triangle in the Poincaré disk model in a symmetric fashion. Then one obvious choice is picking the points in a symmetric fashion as well. Work out a way to characterize that choice without depending on the model, and calculate the distance between these points. – MvG Nov 02 '17 at 07:30