Armed with explicit formulas for null points and null lines, along with their meets and joins, we return to the polarity of Apollonius with which we began this series. Our aim is to establish a fundamental fact that was previously stated without proof: that the dual or polar of a point can be found by two auxiliary (interior) lines and an associated quadrangle of null points. The key point is that the diagonal line formed by the (other) diagonal points of this quadrangle depend only on the original point. Our main tool is an explicit---but lengthy!---- formula for the meet of two interior lines formed by two pairs of null points. As usual we illustrate with a concrete explicit example. CONTENT SUMMARY: pg 1: @00:10 null point, null line, join of null points, meet of null lines F(t1:u1|t2:u2), f(t1:u1|t2:u2); interior line, exterior point pg 2: @04:45 drawing an interior line and exterior point pg 3: @07:09 quadrangle, quadrilateral, Apollonius, polarity pg 4: @08:36 a quadrangle of 4 null points; g function for joins and a meet; formula for the meet of 2 interior lines pg 5: @12:49 quadrangle computation example; important observation about 3 diagonal points; statement of polarity of Appolonius pg 6: @18:40 Nil quadrangle diagonals theorem, proof pg 7: @21:33 calculation showing 3 diagonal points are mutually perpendicular; pg 8: @23:51 homogeneous coordinates to affine coordinates, pole/polar corollary (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.