Rational trigonometry, generalized triangle geometry and four-fold incenter symmetry

This is a seminar talk given to the School of Mathematics and Statistcs, UNSW in April 2013. It describes joint work with Nguyen Le on a generalized triangle geometry.
We begin with an introduction to rational trigonometry--an approach to the subject that is almost purely algebraic, replacing distances and angles with quadrances and spreads and allowing extension to general fields, and indeed to arbitrary bilinear forms. In particular the formulas of rational trigonometry apply to relativistic geometry, including the red and green planar geometries that complement the familiar Euclidean blue geometry. Together these three geometries combine in a lovely way, leading to a subject called chromogeometry.

Triangle geometry is seeing a great revival of interest due to the power of dynamic geometry packages such as GSP, C.a.R and Geogebra, and the online Encyclopedia of Triangle Centers maintained by Clark Kimberling. Here we present a very general approach to this subject, using rational trigonometry and linear transformations to reduce to a standard triangle. This becomes particularly interesting when studying the incenter hierarchy and the natural four-fold symmetry it supports. I report on some interesting concurrences associated to the incenters, Gergonne and Nagel points, Mittenpunkts, Spieker and Bevan points.

Finally I mention a remarkable result on what happens when we consider the blue, red and green incenters of a given triangle--all at the same time!

Video Content:
00:00 Introduction
2:09 Quadrance between points
4:21 Pythagoras theorem
6:16 Blue Pythagoras
6:42 Red Pythagoras
7:33 Green Pythagoras
8:23 Triple quad formula
10:31 Spread between lines
12:06 Spread as a normalized squared determinant
12:30 Laws of affine rational trigonometry
13:33 Triangle geometry and the blue Euler line
15:50 Red Euler line
16:42 Green Euler line
16:58 Omega triangle and Chromogeometry
18:28 Triangle geometry with general bilinear forms
20:08 Standard coordinates
22:30 Standard quadrances and spreads
24:51 The Euler line
26:45 Bilines and Incenter formulas
28:02 Bilines, Incenters, Incircles, contact points
29:39 Gergonne points G
31:44 Nagel points
33:27 Gergonne-Nagel lines and X(69)
35:01 Isogonal and isotomic conjugates
37:45 Gergonne-Nagel lines and X(69)
38:00 Bevan points B
39:31 Mittenpunkts
40:46 Summary
43:40 Summary
44:51 Summary
44:55 Mittenpunkts
45:40 Coloured Incenter Circles
47:58 More info/References
************************

Here are the Insights into Mathematics Playlists:

[njwildberger]

That's a good question. Walter Bloom and I have thought about that, and decided that any triple of bilinear forms obeying the same pleasant three fold symmetries must be conjugate in a natural way to the blue, red and green geometries.

[njwildberger]

No, this is a special tool just designed to study triangle geometry.

[pickeyberry]

Video Content
00:00 Introduction
2:09 Quadrance between points
4:21 Pythagoras theorem
6:16 Blue Pythagoras
6:42 Red Pythagoras
7:33 Green Pythagoras
8:23 Triple quad formula
10:31 Spread between lines
12:06 Spread as a normalized squared determinant
12:30 Laws of affine rational trigonometry
13:33 Triangle geometry and the blue Euler line
15:50 Red Euler line
16:42 Green Euler line
16:58 Omega triangle and Chromogeometry
18:28 Triangle geometry with general bilinear forms
20:08 Standard coordinates
22:30 Standard quadrances and spreads
24:51 The Euler line
26:45 Bilines and Incenter formulas
28:02 Bilines, Incenters, Incircles, contact points
29:39 Gergonne points G
31:44 Nagel points
33:27 Gergonne-Nagel lines and X(69)
35:01 Isogonal and isotomic conjugates
37:45 Gergonne-Nagel lines and X(69)
38:00 Bevan points B
39:31 Mittenpunkts
40:46 Summary
43:40 Summary
44:51 Summary
44:55 Mittenpunkts
45:40 Coloured Incenter Circles
47:58 More info/References

[postbodzapism]

Rewatching this video, I am still astounded by the power of standard coordinates.
I think it would be nice if I have a computer with me so that I can check each calculation of yours. It would be a pleasant experience.
It would also be nice if we can have a purely geometric reasoning of your arguments (esp existence of isogonal / isotomic conjugates) (as opposed to purely computational ones)

[postbodzapism]

As per the chromeogeometry again: do you think they can be used on Riemannian geometry? (after you have a solid foundation on tensor calculus) Using the traditional index notation, for example, Z_ij = vector(Z_i) dot vector(Z_j) by definition. But here the dot product can be changed into the blue, red and green billinear forms...
This would lead us to think about the three billinear forms in the general, "tensorial" setting independent of chosen coordinates, and how to characterize them.

[njwildberger]

No it isn't in general, at least not that I know of.