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From Wikipedia's triangle center article: "Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The Encyclopedia of Triangle Centers is an ever-expanding list of interesting ones."

Hyperbolic Barycentric Coordinates are used in: http://ajmaa.org/searchroot/files/pdf/v6n1/v6i1p18.pdf

What makes a triangle center interesting, and why would hyperbolic triangle centers be interesting ? Wouldn't they just be copies of the euclidean triangle centers ?

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+1 for the interesting paper. –  draks ... Apr 6 '12 at 10:57

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The Encyclopedia of Triangle Centers -- it's only up to 5389 centers at the moment, after decades of searches by dozens of mathematicians using hundreds of computers. The main claim to fame of these 5389 functions is that they are invariant, so that $f(a,b,c) = f(b,c,a) = f(c,b,a)$. This property of invariance is enough to to make the center interesting enough to be listed. In the "uninteresting" example listed at Wikipedia, the same function gives different values depending on how the variables are ordered -- most functions would fit into this category.

Hyperbolic triangle centers are especially interesting when they don't have a planar counterpart.

Centers are weirdly connected to each other. Also take a look at Cubics in the Triangle Plane

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