Euler's Four Point Relation 
Euler's Four Point Relation
by UNSW / N.J. Wildberger
Video Lecture 69 of 72
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Views: 1,563
Date Added: August 2, 2011

Lecture Description

In this video lecture, Prof. N.J. Wildberger Euler's four point relation. 

Euler found the volume of a tetrahedron in terms of the six quadrances of its sides. When the four points lie instead in a plane, the volume is zero, so Euler's formula gives a relation between the six quadrances. We work towards discovering this relation by studying a particular quadrilateral in the plane, and using rational trigonometry to determine the sixth quadrance from the other five.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

Course Index

  1. An Invitation to Geometry: The WildTrig Series
  2. Why Trig is Hard
  3. Quadrance via Pythagoras and Archimedes
  4. Spread, Angles and Astronomy
  5. Five Main Laws of Rational Trigonometry
  6. Applications of Rational Trigonometry
  7. Heron's Formula Viewed Rationally
  8. Solving Triangles With Rational Trigonometry
  9. Centers of Triangles With Rational Trigonometry
  10. The Laws of Proportion for a Triangle
  11. The Laws of Proportion for a Triangle
  12. Geometry of Circles with Rational Trigonometry
  13. Applications of Rational Trig to Surveying
  14. Cartesian Coordinates and Geometry
  15. Why Spreads are Better than Angles
  16. Rational Parameters for Circles
  17. Complex Numbers and Rotations
  18. Rational Trigonometry Quiz 1
  19. Rational Trigonometry: Solutions to Quiz 1
  20. Medians, Altitudes and Vertex Bisectors
  21. Trigonometry With Finite Fields (I)
  22. Trigonometry with Finite Fields (II)
  23. Trigonometry with Finite Fields (III)
  24. Highlights From Triangle Geometry (I)
  25. Highlights From Triangle Geometry (II)
  26. Spread Polynomials
  27. Pentagons and Five-fold Symmetry
  28. Applications of Rational Trig to Surveying (II)
  29. Stewart's Theorem
  30. What Size Ladder Fits Around a Corner?
  31. Trisecting Angles and Hadley's Theorem
  32. Polar Coordinates and Rational Trigonometry
  33. Introduction to Projective Geometry
  34. Projective Geometry and Perspective
  35. Projective Geometry and Homogeneous Coordinates
  36. Lines and Planes in Projective Geometry
  37. Affine Geometry and Barycentric Coordinates
  38. Affine Geometry and Vectors
  39. The Cross Ratio
  40. More About the Cross Ratio
  41. Harmonic Ranges and Pencils
  42. The Fundamental Theorem of Projective Geometry
  43. Conics via Projective Geometry
  44. An Algebraic Framework for Rational Trigonometry (Part I)
  45. An Algebraic Framework for Rational Trigonometry (Part II)
  46. How to Learn Mathematics
  47. Einstein's Special Relativity: An Introduction
  48. Red Geometry (Part I)
  49. Red Geometry (Part II)
  50. Red Geometry (Part III)
  51. Circles in Red Geometry
  52. Green Geometry (Part I)
  53. Green Geometry (Part II)
  54. Pythagorean Triples
  55. An Introduction to Chromogeometry
  56. Chromogeometry and Euler Lines
  57. Chromogeometry and the Omega Triangle
  58. Chromogeometry and Nine-point Circles
  59. Proofs in Chromogeometry
  60. Triangle Spread Rules
  61. Triangle Spread Rules in Action
  62. Acute and Obtuse Triangles
  63. Proofs of the Triangle Spread Rules
  64. Rational Trigonometry Quiz #2
  65. Hints for Solutions to Quiz #2
  66. The 6-7-8 Triangle
  67. Barycentric Coordinates and the 6-7-8 Triangle
  68. Squares in a Pentagon
  69. Trisecting a Right Triangle
  70. Euler's Four Point Relation
  71. What is Geometry Really About?
  72. Determinants in Geometry (Part I)
  73. Determinants in Geometry (Part II)

Course Description

In this course, Prof. N.J. Wildberger gives 72 video lectures on Rational Trigonometry.

This video series on Rational Trigonometry (RT) and related geometry presents a much needed alternative to the traditional tedious and painful subject of trigonometry, which alienates millions of students each year from mathematics. By dispensing with transcendental notions, circular functions and square roots, this new theory gives simpler, faster and more accurate ways to solve a wide variety of engineering, surveying, physics and geometry problems, essentially only with high school algebra (that's right, calculators or trig tables are not required). RT is also a much more satisfying and logical way to introduce young people to the beauty and elegance of geometry, and teaches them that mathematics should, first and foremost, always make sense. Prepare to depart on a modern adventure, in the spirit of the ancient Greeks! Assoc Prof N J Wildberger is the author of the first book on this subject, 'Divine Proportions: Rational Trigonometry to Universal Geometry'. He is also and innovative and highly regarded teacher in the School of Mathematics and Statistics at UNSW.


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