Rational Trigonometry, Generalized Triangle Geometry and Four-Fold Incenter Symmetry

Explore a seminar talk on rational trigonometry and generalized triangle geometry, delving into an algebraic approach that replaces traditional concepts with quadrances and spreads. Learn about the application of rational trigonometry to relativistic geometry and chromogeometry. Discover a general approach to triangle geometry using linear transformations and standard triangles. Examine the incenter hierarchy, its four-fold symmetry, and interesting concurrences associated with various triangle centers. Investigate the blue, red, and green Pythagoras theorems, Euler lines, and incenter circles. Gain insights into bilines, Gergonne points, Nagel points, and Mittenpunkts. Uncover the connections between different geometric concepts and their applications in this comprehensive exploration of advanced trigonometry and triangle geometry.

Introduction
Quadrance between points
Pythagoras theorem
Blue Pythagoras
Red Pythagoras
Green Pythagoras
Triple quad formula
Spread between lines
Spread as a normalized squared determinant
Laws of affine rational trigonometry
Triangle geometry and the blue Euler line
Red Euler line
Green Euler line
Omega triangle and Chromogeometry
Triangle geometry with general bilinear forms
Standard coordinates
Standard quadrances and spreads
The Euler line
Bilines and Incenter formulas
Bilines, Incenters, Incircles, contact points
Gergonne points G
Nagel points
Gergonne-Nagel lines and X69
Isogonal and isotomic conjugates
Gergonne-Nagel lines and X69
Bevan points B
Mittenpunkts
Summary
Summary
Summary
Mittenpunkts
Coloured Incenter Circles
More info/References