## Universal Hyperbolic Triangle Geometry

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• Dear fellow geometers, I would like to alert you to a new form of hyperbolic triangle geometry, which greatly enriches the subject. It is based on Universal
Message 1 of 3 , Feb 23, 2011
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Dear fellow geometers,

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.

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.

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

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

and also in more detail in my ArXiV paper at

http://arxiv.org/abs/0909.1377

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

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.

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

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.

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

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.

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.

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

All the best,

N J Wildberger (UNSW)

n.wildberger@...
• Further to this: I have now decided to call this most distinguished line in universal hyperbolic geometry the ortho-axis rather than the orthopole line .
Message 2 of 3 , May 17, 2011
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Further to this: I have now decided to call this most distinguished line in universal hyperbolic geometry the "ortho-axis" rather than the "orthopole line". The definition of this line can be found in my YouTube video "UnivHypGeom10: Orthocenters exist!" which is at

This is part of my new series of videos on Universal Hyperbolic Geometry, giving an elementary introduction to the subject.

All the best,

Norman Wildberger

--- In Hyacinthos@yahoogroups.com, "njwildberger" <njwildberger@...> wrote:
>
> Dear fellow geometers,
>
> 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.
>
> 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.
>
> UHG is described in my survey paper in the recent edition of KoG:
>
> http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=94152
>
> and also in more detail in my ArXiV paper at
>
> http://arxiv.org/abs/0909.1377
>
> Triangle geometry in this setting complements and sheds light on the Euclidean theory. Here are some interesting aspects:
>
> 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.
>
> 2. A complete duality between points and lines, and between the metrical notions of quadrance and spread. This is a purely projective theory.
>
> 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.
>
> 4. The theory works over a general field, in particular over the rational numbers, or a finite field.
>
> 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.
>
> 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.
>
> I am happy to answer questions/ take comments, this post probably as good a way as any.
>
> All the best,
>
> N J Wildberger (UNSW)
>
> n.wildberger@...
>
• Dear Geometers, I am compiling an initial work on Triangle Geometry in the Universal Hyperbolic Geometry (UHG) setting and would like to float some ideas on
Message 3 of 3 , Oct 30, 2011
• 0 Attachment
Dear Geometers,

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:
>
> Dear fellow geometers,
>
> 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.
>
> 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.
>
> UHG is described in my survey paper in the recent edition of KoG:
>
> http://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=94152
>
> and also in more detail in my ArXiV paper at
>
> http://arxiv.org/abs/0909.1377
>
> Triangle geometry in this setting complements and sheds light on the Euclidean theory. Here are some interesting aspects:
>
> 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.
>
> 2. A complete duality between points and lines, and between the metrical notions of quadrance and spread. This is a purely projective theory.
>
> 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.
>
> 4. The theory works over a general field, in particular over the rational numbers, or a finite field.
>
> 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.
>
> 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.
>
> I am happy to answer questions/ take comments, this post probably as good a way as any.
>
> All the best,
>
> N J Wildberger (UNSW)
>
> n.wildberger@...
>
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