I am compiling an initial work on Triangle Geometry in the Universal Hyperbolic Geometry (UHG) setting and would like to float some ideas on terminology past you: basically the use of the terms: Apollonian point and Centrian, but a few other modifications too if you will allow me.

Note the use of capital letters in the words I am about to define. Fix a triangle say ABC, call A,B,C its Points, and AB,AC,BC its Lines.

Suppose that a triangle has 6 Bilines (you would say angle bisectors), 2 through each Point (this is not automatic in UHG). These Bilines meet three at a time at 4 Incenters (you would say one incenter, 3 excenters). Each Biline meets the opposite Line at an Apollonian point. The Apollonian points are collinear three at a time on 4 Centrians.

Is there a standard name for this last result? I appreciate any comments.

Regards,

N J Wildberger (UNSW)

--- In Hyacinthos@yahoogroups.com, "njwildberger" <njwildberger@...> wrote:

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> Dear fellow geometers,

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> I would like to alert you to a new form of hyperbolic triangle geometry, which greatly enriches the subject. It is based on Universal Hyperbolic Geometry (UHG), a purely algebraic approach devoid of transcendental functions like cosh, tanh, log, which parallels Rational Trigonometry (RT) in the plane.

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> If you haven't heard of RT, please have a look at my web pages at http://web.maths.unsw.edu.au/~norman/Rational1.htm or check out my YouTube video series called WildTrig. RT is very important for triangle geometry, and will clarify many things which you already know.

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> UHG is described in my survey paper in the recent edition of KoG:

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> http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=94152

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> and also in more detail in my ArXiV paper at

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> http://arxiv.org/abs/0909.1377

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> Triangle geometry in this setting complements and sheds light on the Euclidean theory. Here are some interesting aspects:

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> 1. Many common features to Euclidean TG: orthocenters, circumcenters, incenters, isogonal and isotomic conjugates, barycentric coords, Gergonne, Nagel, symmedian points, isodynamic points, cyclocevian conjugates, Brocard points, Soddy line, etc. But also many of the familiar constructions seem to have no obvious parallel: the Euler line, nine-point circle, Fermat points, Feuerbach thm, Napolean and Morley thms etc.

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> 2. A complete duality between points and lines, and between the metrical notions of quadrance and spread. This is a purely projective theory.

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> 3. A VERY distinguished new line with no Euclidean analog--- what I call the orthopole line--- which contains a good fraction of notable new points. Yes, there are MANY new points, some very interesting.

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> 4. The theory works over a general field, in particular over the rational numbers, or a finite field.

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> 5. There is a kaleidescopic aspect to the subject which makes if combinatorially richer than the Euclidean theory. There are interesting connections with Lie theory and combinatorics, also a kind of algebraic geometry involving spaces of conics.

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> I can almost guarantee that if you investigate this new development, you will love it. It is really beautiful. I hope to have the third paper in the series (on Triangle Geometry) finished in the next year, and I'd like to discuss some of this with you as I go, if there is interest.

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> I am happy to answer questions/ take comments, this post probably as good a way as any.

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> All the best,

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> N J Wildberger (UNSW)

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> n.wildberger@...

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