Rational Trigonometry

Video Lectures

Displaying all 73 video lectures.
Lecture 1
An Invitation to Geometry: The WildTrig Series
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An Invitation to Geometry: The WildTrig Series
In this video lecture, Prof. N.J. Wildberger introduces the WildTrig series, inviting you to learn a new approach to geometry and trigonometry. This series will give a careful introduction to rational trigonometry and universal geometry---valid over a general field.
Lecture 2
Why Trig is Hard
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Why Trig is Hard
In this video lecture, Prof. N.J. Wildberger talks about why trigonometry is hard.

The usual trigonometry is overly complicated, inaccurate and logically dubious. This is the first of a series that shows you a better way---rational trigonometry. Rational trigonometry replaces distance and angle with more algebraic notions called quadrance and spread. All of those complicated formulas become much simpler, you don't need a calculator any more, and the theory extends Euclidean geometry to arbitrary fields. What mathematics teacher or student could ask for more?

Assoc. Prof. N J Wildberger from UNSW is also the creator of the MathFoundations series, the WildLinAlg series, and a more advanced video series on Algebraic Topology. 
Lecture 3
Quadrance via Pythagoras and Archimedes
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Quadrance via Pythagoras and Archimedes
In this video lecture, Prof. N.J. Wildberger shows how the basic notion of rational trigonometry---quadrance---arises from the geometry of the ancient Greeks. The little-known sister theorem to Pythagoras features prominently, and is closely related to a theorem of Archimedes.
Lecture 4
Spread, Angles and Astronomy
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Spread, Angles and Astronomy
In this video lecture, Prof. N.J. Wildberger talks about spread, angles and astronomy. Angles have their origin in astronomy and spherical trigonometry. Here we introduce the rational alternative, called spread, and give examples from ISO paper sizes to the faces of a dodecahedron.
Lecture 5
Five Main Laws of Rational Trigonometry
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Five Main Laws of Rational Trigonometry
In this video lecture, Prof. N.J. Wildberger talks about five main laws of rational trigonometry. We derive from first principles the main laws of rational trigonometry, using the concepts of quadrance and spread to replace the usual distance and angle. Most everything works out much simpler.
Lecture 6
Applications of Rational Trigonometry
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Applications of Rational Trigonometry
In this video lecture, Prof. N.J. Wildberger applies rational trigonometry to solve four examples of practical problems, concerning a flagpole, a ladder, a kite and the distance from a point to a line.
Lecture 7
Heron's Formula Viewed Rationally
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Heron's Formula Viewed Rationally
Heron's formula, originally due to Archimedes, is here recast in a simpler and more natural form. In this video lecture, Prof. N.J. Wildberger proves it, using one of the basic laws of rational trigonometry.
Lecture 8
Solving Triangles With Rational Trigonometry
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Solving Triangles With Rational Trigonometry
In this video lecture, Prof. N.J. Wildberger shows how to solve triangles using the framework of rational trigonometry with quadrances and spreads replacing distance and angle.
Lecture 9
Centers of Triangles With Rational Trigonometry
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Centers of Triangles With Rational Trigonometry
Lecture 9
The Laws of Proportion for a Triangle
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The Laws of Proportion for a Triangle
Lecture 10
The Laws of Proportion for a Triangle
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The Laws of Proportion for a Triangle
Lecture 11
Geometry of Circles with Rational Trigonometry
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Geometry of Circles with Rational Trigonometry
Lecture 12
Applications of Rational Trig to Surveying
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Applications of Rational Trig to Surveying
Lecture 13
Cartesian Coordinates and Geometry
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Cartesian Coordinates and Geometry
Lecture 14
Why Spreads are Better than Angles
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Why Spreads are Better than Angles
Lecture 15
Rational Parameters for Circles
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Rational Parameters for Circles
Lecture 16
Complex Numbers and Rotations
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Complex Numbers and Rotations
Lecture 17
Rational Trigonometry Quiz 1
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Rational Trigonometry Quiz 1
Lecture 18
Rational Trigonometry: Solutions to Quiz 1
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Rational Trigonometry: Solutions to Quiz 1
Lecture 19
Medians, Altitudes and Vertex Bisectors
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Medians, Altitudes and Vertex Bisectors
Lecture 20
Trigonometry With Finite Fields (I)
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Trigonometry With Finite Fields (I)
Lecture 21
Trigonometry with Finite Fields (II)
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Trigonometry with Finite Fields (II)
Lecture 22
Trigonometry with Finite Fields (III)
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Trigonometry with Finite Fields (III)
Lecture 23
Highlights From Triangle Geometry (I)
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Highlights From Triangle Geometry (I)
Lecture 24
Highlights From Triangle Geometry (II)
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Highlights From Triangle Geometry (II)
Lecture 25
Spread Polynomials
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Spread Polynomials
Lecture 26
Pentagons and Five-fold Symmetry
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Pentagons and Five-fold Symmetry
Lecture 27
Applications of Rational Trig to Surveying (II)
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Applications of Rational Trig to Surveying (II)
Lecture 28
Stewart's Theorem
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Stewart's Theorem
Lecture 29
What Size Ladder Fits Around a Corner?
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What Size Ladder Fits Around a Corner?
Lecture 30
Trisecting Angles and Hadley's Theorem
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Trisecting Angles and Hadley's Theorem
Lecture 31
Polar Coordinates and Rational Trigonometry
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Polar Coordinates and Rational Trigonometry
Lecture 32
Introduction to Projective Geometry
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Introduction to Projective Geometry
Lecture 33
Projective Geometry and Perspective
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Projective Geometry and Perspective
Lecture 34
Projective Geometry and Homogeneous Coordinates
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Projective Geometry and Homogeneous Coordinates
Lecture 35
Lines and Planes in Projective Geometry
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Lines and Planes in Projective Geometry
In this video lecture, Prof. N.J. Wildberger talks about lines and planes in projective geometry. 

How to think about both projective points and projective lines via lines and planes in 3D geometry. Also we discuss some basic facts about 3D geometry, relating perpendicularity and quadrances.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 36
Affine Geometry and Barycentric Coordinates
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Affine Geometry and Barycentric Coordinates
In this video lecture, Prof. N.J. Wildberger talks about affine geometry and barycentric coordinates. 

Affine geometry is the geometry of parallel lines. Using parallelism, we show how to construct a ruled line, how to find the midpoint of a segment, and divide a segment into a given ratio. We connect this to Archimedes law of the lever, and then extend to barycentric coordinates with respect to a triangle.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 37
Affine Geometry and Vectors
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Affine Geometry and Vectors
In this video lecture, Prof. N.J. Wildberger talks about affine geometry and vectors. 

Using vectors, we define parallelograms, discuss affine combinations, and show how to derive barycentric coordinates without any notion of weights.This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 38
The Cross Ratio
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The Cross Ratio
In this video lecture, Prof. N.J. Wildberger talks about the cross ratio. 

The cross ratio is the most important invariant in projective geometry, and plays a key role in hyperbolic geometry. We introduce it here using vectors, that is in the framework of affine geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 39
More About the Cross Ratio
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More About the Cross Ratio
In this video lecture, Prof. N.J. Wildberger talks about cross ratio. 

We extend the cross ratio from four collinear points to four concurrent lines, and introduce the special cases of harmonic ranges and harmonic pencils.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 40
Harmonic Ranges and Pencils
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Harmonic Ranges and Pencils
In this video lecture, Prof. N.J. Wildberger talks about Harmonic Ranges and Pencils. 

Four points on a line with a cross ratio of -1 form a harmonic range. Four lines through a point with a cross ratio of -1 form a harmonic pencil. These two notions are intimately linked, and related naturally to quadrangles and quadrilaterals.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 41
The Fundamental Theorem of Projective Geometry
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The Fundamental Theorem of Projective Geometry
In this video lecture, Prof. N.J. Wildberger talks about the Fundamental Theorem of Projective Geometry. 

The fundamental theorem of projective geometry states that any four planar non-collinear points (a quadrangle) can be sent to any quadrangle via a projectivity, that is a sequence of perspectivities. We prove this by first establishing the simpler one-dimensional case of three points on a projective line.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry. 
Lecture 42
Conics via Projective Geometry
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Conics via Projective Geometry
In this video lecture, Prof. N.J. Wildberger talks about conics via projective geometry. 

Conics, such as circles, ellipses, hyperbolas and parabolas, can be defined purely within projective geometry, as realized by the nineteenth century German mathematician Steiner. This is done by using projectivities. There are essentially two dual constructions, one giving a line conic, the other a point conic. We illustrate using The Geometer's Sketchpad, a useful software program for students of geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry. 
Lecture 43
An Algebraic Framework for Rational Trigonometry (Part I)
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An Algebraic Framework for Rational Trigonometry (Part I)
In this video lecture, Prof. N.J. Wildberger talks about rational trigonometry, which can be developed purely algebraically, without any pictures. This video reminds you of the basic concepts of quadrance and spread and their definitions in terms of coordinates.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 44
An Algebraic Framework for Rational Trigonometry (Part II)
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An Algebraic Framework for Rational Trigonometry (Part II)
In this video lecture, Prof. N.J. Wildberger talks about the most powerful law in geometry: the Cross law, the rational analog of the Cosine law. It includes as special cases Pythagoras' theorem and the Triple quad formula. Here we sketch a purely algebraic derivation of the Cross law, and then how the other four main laws of rational trigonometry follow from it.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 45
How to Learn Mathematics
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How to Learn Mathematics
In this video lecture, Prof. N.J. Wildberger provides a few thoughts on how to learn mathematics, that should also be relevant for this course in Rational Trigonometry and Geometry. The basic idea: 'Mathematics is a landscape'. So learning mathematics is not that different from becoming knowledgable about an unfamiliar city or terrain. This is a short break from our usual thread of rational trigonometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 46
Einstein's Special Relativity: An Introduction
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Einstein's Special Relativity: An Introduction
In this video lecture, Prof. N.J. Wildberger talks about Einstein's Special Relativity. 

Einstein's special theory of relativity (1905) was recast by Minkowski in terms of the geometry of a four dimensional spacetime. This video gives an introduction to this idea, motivating our study of two dimensional relativistic (red) geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 47
Red Geometry (Part I)
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Red Geometry (Part I)
In this video lecture, Prof. N.J. Wildberger talks about red geometry. 

Red geometry is a two dimensional relativistic geometry in the spirit of rational trigonometry, using variants of the usual quadrance and spread. The usual grid plane is still the arena in which this geometry lives, but the notion of perpendicularity is new. A wonderful new world of geometry emerges from just a twist of the old definitions. 

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 48
Red Geometry (Part II)
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Red Geometry (Part II)
In this video lecture, Prof. N.J. Wildberger talks about Red Geometry. 

We give some examples of calculating quadrances and spreads in red geometry, and illustrating some of the usual laws of rational trigonometry. The concurrence of the red circumcenter, centroid and orthocenter on the 'red Euler line' is shown in a special case.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 49
Red Geometry (Part III)
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Red Geometry (Part III)
In this video lecture, Prof. N.J. Wildberger talks about Red Geometry.

Remarkably, the main laws of trigonometry in the relativistic (red) setting are exactly the same as those in the Euclidean setting. To establish this, we show first that the Cross law---the most powerful law---holds in red geometry. Then the others, such as Pythagoras, the Triple quad formula, the Spread law and the Triple spread formula, follow much as they do in ordinary (blue) geometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 50
Circles in Red Geometry
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Circles in Red Geometry
In this video lecture, Prof. N.J. Wildberger talks about circles in red geometry.  

A circle in red geometry looks like a rectangular hyperbola with asymptotes parallel to the lines of slope one and minus one. But in the red setting, such a `red' circle still satisfies many of the properties that we are used to for a circle. So this video shares some of the theorems of WildTrig 10, but in a completely different context.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 51
Green Geometry (Part I)
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Green Geometry (Part I)
In this video lecture, Prof. N.J. Wildberger talks about Green Geometry.

There is a second form of planar relativistic geometry, here called green geometry, which plays a similar role to red geometry and is closely related to it. This video introduces this geometry via quadrance and perpendicularity.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 52
Green Geometry (Part II)
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Green Geometry (Part II)
In this video lecture, Prof. N.J. Wildberger talks about Green Geometry. 

We show that the laws of trigonometry in the relativistic green setting are exactly the same as for Euclidean geometry, and the same as for the red relativistic geometry. We also draw some pictures of `green circles', which are rectangular hyperbolas with asymptotes the coordinate axes, and give as an exercise the equal products theorem.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 53
Pythagorean Triples
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Pythagorean Triples
In this video lecture, Prof. N.J. Wildberger talks about Pythagorean Triples. 

A Pythagorean triple consists of three natural numbers x, and z satisfying x^2+y^2=z^2 . By Pythagoras' theorem, this means these three numbers are the sides of a right triangle. Euclid knew how to solve this equation, and the solution involves three expressions which form also the basis of chromogeometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 54
An Introduction to Chromogeometry
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An Introduction to Chromogeometry
In this video lecture, Prof. N.J. Wildberger gives an introduction to Chromogeometry. 

Chromogeometry is a new three-fold symmetry in planar geometry, that brings together the usual Euclidean geometry (here called `blue') with two relativistic geometries (called `red' and `green'). There are two basic principles: that the three geometries are essentially alike, and that they all fit together to form a harmonious whole.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 55
Chromogeometry and Euler Lines
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Chromogeometry and Euler Lines
In this video lecture, Prof. N.J. Wildberger talks about chromogeometry and Euler lines. 

Chromogeometry allows many features of Euclidean geometry to be widely generalized and enriched. Here we look at the classical Euler line of a triangle, passing through the orthocenter, centroid and circumcenter also from two relativistic geometries. Then we combine all three pictures to produce the remarkable Omega triangle.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 56
Chromogeometry and the Omega Triangle
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Chromogeometry and the Omega Triangle
In this video lecture, Prof. N.J. Wildberger talks about chromogeometry and the omega triangle. 

Chromogeometry allows us to study not just one, but three Euler lines of a triangle. One for each of the three fundamental planar geometries: Euclidean (blue) and the two relativistic geometries called red and green. These Euler lines turn out to be nothing but the medians of the triangle of orthocenters of the original triangle, called the Omega triangle. Many interesting features arise relating orthocenters, circumcenters and nine-point centers of different colours.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 57
Chromogeometry and Nine-point Circles
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Chromogeometry and Nine-point Circles
In this video lecture, Prof. N.J. Wildberger talks about chromogeometry and nine-point circles. 

Chromogeometry allows us to consider not just the usual nine-point circle of a triangle, but also a red and a green nine-point circle, which are in fact rectangular hyperbolas. This video uses The Geometer's Sketchpad to illustrate the various special points to be found on these nine-point circles, and how they interact.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 58
Proofs in Chromogeometry
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Proofs in Chromogeometry
In this video lecture, Prof. N.J. Wildberger outlines how we approach proofs in chromogeometry. The emphasis is on algebraic relations and polynomial identities. A convenient shorthand notation for anti-symmetric polynomials is introduced, and formulas for the orthocenters and circumceneters of a given triangle, in all three coloured geometries---blue, red and green --- are given.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 59
Triangle Spread Rules
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Triangle Spread Rules
In this video lecture, Prof. N.J. Wildberger talks about triangle spread rules. The three spreads of a triangle satisfy the Triple spread formula, and given two such spreads this gives a quadratic equation for the third. The triangle spread rules give information about which of the two solutions of this quadratic equation to take in particular circumstances. The notion of sector is introduced, along with notions of acute and obtuse.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 60
Triangle Spread Rules in Action
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Triangle Spread Rules in Action
In this video lecture, Prof. N.J. Wildberger looks at a quadrilateral problem using rational trigonometry and the triangle spread rules. The triangle spread rules allow us to determine which of two solutions of the Cross law or Triple spread formula is appropriate in many situations.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 61
Acute and Obtuse Triangles
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Acute and Obtuse Triangles
In this video lecture, Prof. N.J. Wildberger talks about acute and obtuse triangles. 

We define acute and obtuse triangles, and give inequalities for acuteness in terms of the spreads of the triangle. Then we look at the circumcenter of the triangle, and express it as an affine combination of the vertices. The Triangle spread rules play a role.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 62
Proofs of the Triangle Spread Rules
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Proofs of the Triangle Spread Rules
In this video lecture, Prof. N.J. Wildberger gives proofs of the two triangle spread rules, which tell us how to choose the third spread of a triangle given two spreads. The arguments are somewhat complicated, involving some rather magical identities.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 63
Rational Trigonometry Quiz #2
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Rational Trigonometry Quiz #2
In this video lecture, Prof. N.J. Wildberger gives a second quiz on rational trigonometry, giving four challenging trigonometric problems that can also be tackled with ordinary trigonometry (probably with greater effort and less success, but you can try and see). One problem is on vertex bisectors and medians of a particular 6-7-8 triangle, another on squares built inside a pentagon, another on trisecting a right triangle, and the fourth on a flexible quadrilateral. These problems should be a fun challenge. Warning: they are not easy.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 64
Hints for Solutions to Quiz #2
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Hints for Solutions to Quiz #2
In this video lecture, Prof. N.J. Wildberger gives hints for the four problems of Quiz #2, one concerning the 6-7-8 triangle, another about squares inscribed in the sides of a pentagon, another on trisecting a right triangle, and the last---a harder problem--on finding the diagonals of a quadrilateral given the sides and the angle/spread made by the diagonals.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 65
The 6-7-8 Triangle
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The 6-7-8 Triangle
In this video lecture, Prof. N.J. Wildberger talks about the 6-7-8 triangle. 

We look at a problem involving a triangle with side lengths 6,7 and 8. We need to find a distance from one vertex to the meet of a median and a vertex (or angle) bisector. In this video we give a solution involving straigthforward triangle by triangle analysis using rational trigonometry.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 66
Barycentric Coordinates and the 6-7-8 Triangle
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Barycentric Coordinates and the 6-7-8 Triangle
In this video lecture, Prof. N.J. Wildberger talks about Barycentric Coordinates and the 6-7-8 Triangle. 

Here we present another solution to the 6-7-8 triangle problem, this time using barycentric coorindates. This gives us a chance to apply the idea of assigning weights to the vertices of a triangle to determine various ratios formed by lines from the vertices meeting at a point.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 67
Squares in a Pentagon
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Squares in a Pentagon
In this video lecture, Prof. N.J. Wildberger talks about Squares in a Pentagon. 

We use rational trigonometry to solve a problem concerning a pentagon with inscribed touching squares---the question being to find the ratio of the sides of the pentagon to the squares, and also the ratios of two circles, one circumscribing the pentagon, the other inscribed in the squares.

The Golden spreads of WildTrig 25 (Lecture 26: Pentagons and Five-fold Symmetry) play a key role, as they do for many problems with five fold symmetry. This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 68
Trisecting a Right Triangle
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Trisecting a Right Triangle
In this video lecture, Prof. N.J. Wildberger talks about trisecting a right triangle. 

We analyse the third problem from our Quiz #2 (Lecture 63: Rational Trigonometry Quiz #2). This problem studies a right triangle which has one vertex trisected, and for which we know two of the three lengths into which the opposite side has been divided. We are asked to find the length of the third segment.The second and third spread polynomials figure prominently in the solution.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 69
Euler's Four Point Relation
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Euler's Four Point Relation
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.
Lecture 70
What is Geometry Really About?
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What is Geometry Really About?
In this video lecture, Prof. N.J. Wildberger raises the question: What is geometry really about?

Is it pictures or visual patterns? Or constructions? Or simply theorems? In this lecture, we suggest that geometry is, at its heart, actually something else. We will also have a look at a lovely result for four collinear points, generalizing the Triple quad formula, and related to Brahmagupta's area formula for cyclic quadrilaterals.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 71
Determinants in Geometry (Part I)
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Determinants in Geometry (Part I)
In this video lecture, Prof. N.J. Wildberger determinants in geometry. 

We introduce the determinant of a square matrix, giving explicit formulas for the 1,2,3 and 4 dim cases. The general pattern is also described. Then we give two important examples, one relating to the Triple quad formula and Archimedes' function, and the other to Euler's function, involved in Euler's Four point relation.

This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.
Lecture 72
Determinants in Geometry (Part II)
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Determinants in Geometry (Part II)
In this video lecture, Prof. N.J. Wildberger talks about determinants in geometry. The determinant of a (square) matrix is the most important function in mathematics, appearing in almost every aspect of the subject. Here we look at basic properties of the determinant, illustrated with 2x2 and 3x3 examples. In particular we relate the most important property with matrix multiplication, which is explained.

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