The Cross law is the fourth of the four main laws of trigonometry in the hyperbolic setting. It is also the most complicated, and the most powerful law. This video shows how we can prove it with the help of a remarkable polynomial identity. We also give an application to the relation between the quadrance and spread of an equilateral triangle. CONTENT SUMMARY: pg 1: @00:11 Cross law in Euclidean R.T. review; Cross law in UHG; Cross Dual law pg 2: @06:43 Reluctant exposure to classical hyperbolic geometry; exercise 28.1 @09:25 ; observation - Euclidean cross law as a limiting case @11:02 pg 3: @12:46 Cross law (hyperbolic version); proof; heavenly assistance 16:06 pg 4: @16:21 proof continued; using a computer (Scientific Workplace) to verify an identity @21:41 ; proof complete @26:47 pg 5: @28:22 A pleasant consequence of the cross law; Equilateral triangle theorem; proof; exercise 28-2 @31:24 pg 6: @31:48 exercises 28.3-5 (THANKS to EmptySpaceEnterprise)

1. Introduction to Universal Hyperbolic Geometry
2. Apollonius and Polarity
3. Apollonius and Harmonic Conjugates
4. Pappus' Theorem and the Cross Ratio
5. First Steps in Hyperbolic Geometry
6. The Circle and Cartesian Coordinates
7. Duality, Quadrance and Spread in Cartesian Coordinates
8. The Circle and Projective Homogeneous Coordinates
9. The Circle and Projective Homogeneous Coordinates II
10. Computations with Homogeneous Coordinates
11. Duality and Perpendicularity
12. Existence of Orthocenters
13. Theorems using Perpendicularity
14. Null Points and Null Lines
15. Apollonius and Polarity Revisited
16. Reflections in Hyperbolic Geometry
17. Reflections and Projective Linear Algebra
18. Midpoints and Bisectors
19. Medians, Midlines, Centroids and Circumcenters
20. Parallels and the Double Triangle
21. The J function, sl(2) and the Jacobi identity
22. Pure and Applied Geometry: understanding the continuum
23. Quadrance and Spread
24. Pythagoras' Theorem in Universal Hyperbolic Geometry
25. The Triple Quad Formula in Universal Hyperbolic Geometry
26. Visualizing Quadrance with Circles
27. Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry
28. Trigonometric Laws in Hyperbolic Geometry using Geometer's Sketchpad
29. The Spread Law in Universal Hyperbolic Geometry
30. The Cross Law in Universal Hyperbolic Geometry
31. Thales' Theorem, Right Triangles and Napier's Rules
32. Isosceles Triangles in Hyperbolic Geometry
33. Menelaus, Ceva and the Laws of Proportion
34. Trigonometric Dual Laws and the Parallax Formula
35. Introduction to Spherical and Elliptic Geometries
36. Introduction to Spherical and Elliptic Geometries II
37. Areas and Volumes for a Sphere
38. Classical Spherical Trigonometry
39. Perpendicularity, Polarity and Duality on a Sphere
40. Parametrizing and Projecting a Sphere
41. Rational Trigonometry: An Overview
42. Rational Trigonometry in Three Dimensions

This is a complete and relatively elementary course explaining a new, simpler and more elegant theory of non-Euclidean geometry; in particular hyperbolic geometry. It is a purely algebraic approach which avoids transcendental functions like log, sin, tanh etc, relying instead on high school algebra and quadratic equations. The theory is more general, extending beyond the null circle, and connects naturally to Einstein's special theory of relativity. This course is meant for mathematics majors, bright high school students, high school teachers, engineers, scientists, and others with an interest in mathematics and some basic algebraic skills. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry. Look up for his course in order to familiarize with this new development. He also has recorded very organized and detailed lecture series on Algebraic Topology, History of Mathematic, Linear Algebra, and more.