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## Projective Geometry

I have been wanting to do this for quite some time, and GeoGebra provides a vehicle to do this! I have studied bits and pieces of H.S.M Coxeter's Projective Geometry for the last couple of years. As I go back over it again, I will post dynamic sketches for each section of the text. This process will help me learn it again, and perhaps will help you, too!

## Chapter 1: Introduction

### 1.4 The Simplest Geometric Objects

### 1.5 Projectivities

- The simplest correspondence between a range and a pencil
- Combining correspondences to make more sophisticated ones.

### 1.6 Perspectivities

- The product of two elementary correspondences is a Perspectivity
- Perspectivities as a Function. When you know how A, B, and C get mapped to A", B" and C", you can construct the image of any point X. Play the construction with this!
- Here is the Dual of the above construction.
- THEOREM: Any four collinear points can be interchanged in pairs by a projectivity.

## Chapter 2: Triangles and Quadrangles

### 2.1 Axioms

- There is a point and a line that are not incident.
- Every line is incident with at least three points.
- Any two distinct points are incident with just one line.
- If A,B,C,D are four distinct points, and AB intersects CD, then AC intersects BD.
- If ABC is a plane, there exists a point not on the plane.
- Any two planes have at least two common points.
- The diagonal points of a Complete Quadrilateral are never collinear.
- If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point on the line.

### 2.2 Consequences of the Axioms

- The Diagonal Triangle of a complete quadrilateral is a consequence of the diagonal points never being collinear.

### 2.3 Perspective Triangles

**THEOREM:**If two triangles are perspective from a line they are perspective from a point.**Desargues' Theorem:**If two triangles are perspective from a point they are perspective from a line.- Perspective Triangles
- Three Perspective Triangles

### 2.4 Quadrangular Sets

- A Quadrangular Set is the section of a complete quadrilateral by any line that does not pass through a vertex.
- THEOREM: Each point of a quadrangular set is uniquely determined by the other five!
- An Illustration of the Theorem

### 2.5 Harmonic Sets

Harmonic Sets are a special case of Quadrangular Sets when line g is through two diagonal points of the quadrangle.

## Chapter 3: The Principal of Duality

### 3.2 The Desargues Configuration

### 3.3 The Invariance of the Harmonic Relation

- Three collinear points A, B, and C, determine a fourth point D, the harmonic conjugate of B with respect to A and C. Likewise, three concurrent lines determine a fourth harmonic conjugate line.
- Any section of four harmonic lines, by a line not passing thorugh the point of concurrence, produces four harmonic points, as illustrated here
- A harmonic set of points is projected from any point not on the line by a set of harmonic lines, as illustrated here.

### 3.4 Trilinear Polarity

- Start with a triangle ABC and a point D. Construct the Cevian triangle of the point. The two triangles are in perspective with the point D being the center. Trilinear Pole with polar
- Constructing a Trilinear Polar
- Constructing a Trilinear Pole

### 3.5 Harmonic Nets

A point D is said to be harmonically related to three distinct collinear points A, B, and C if P forms a harmonic set with A, B, and C in any order. The set of all points P is called a * harmonic net* or a

*net of rationality*. Restricting ourselves to A, B, and C, we will have three other points in the harmonic net. However, you could continue this process, using B, C, and P, or A, C, and P.

- Go ahead and play around with this Harmonic Tool. You can discover alot of things about harmonic sets and nets by using this!

## Chapter 4: The Fundamental Theorem and Pappus' Theorem

### 4.1 How Three Pairs Determine a Projectivity

**The Fundamental Theorem of Projective Geometry**

A projectivity is completely determined when three collinear points and the corresponding collinear points are given.

You can replace either one or both of the "three collinear points" with "three concurrent lines" and still have a true statement.

- Constructing the Mapping for points on the same line and points on different lines.

## Chapter 5: One-Dimensional Projectivities

### 5.1 Superposed Ranges

### 5.2 Parabolic Projectivities

**THEOREM:** If a projectivity mapping ABW to BCW forms a harmonic range (AC)(BW), then the mapping is parabolic. Otherwise, hyperbolic.