# Chris Sangwin's GeoGebra page

The following pages contain interesting geometry illustrated with GeoGebra.

## Classical Geometry

1. A circle theorem
This worksheet allows you to experiment with one classical circle theorem.
2. Trisect an angle I
One attempt to divide an arbitrary angle into three equal parts, which does not work!
3. Trisect an angle II
Another attempt to divide an arbitrary angle into three equal parts.
4. Pascal's Theorem
Take six points on a conic section. The intersection of lines joining them are colinear.
5. Tangents to a conic section
Using Pascal's Theorem we can construct the tangent to a conic through a point on the conic.
6. Morley's Theorem
Inside every triangle is an equilateral triangle trying to escape!
7. The nine point circle
This construction shows the "nine point" circle and Euler line.
8. Tangents to two circles
This construction shows how to construct lines tangent to two circles.

## Problems

1. Concurent circles I
Why are these three circles concurrent?
2. Concurent circles II
Why are these four circles concurrent?
3. Touching circles
How do you construct three circles which touch?
4. Circle, segment and lines
A segment moves around a circle......

## Complex numbers

1. Squaring complex numbers
This demonstrates what happens to lines and circles when points are multiplied by themselves.
2. 1/z
This demonstrates what happens to lines and circles when we take the reciprocal of points.
3. Mobius Transformation
This demonstrates the Mobius transformation.

## Mechanisms

A classical mechanism, illustrated dynamically.

## 3D geometry

The harmonograph, related to Lissajous figures. Anaglyph figures use red and green glasses. While this is a radical way to see three dimensions, it is effective.

## Chaos

1. Cobwebbing
This illustrates the procedure of "cobwebbing" - a graphical method of looking at the long term behaviour of one dimensional maps.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)
2. The logistic map
This worksheet shows the logistic map (Dark blue) f(x)=a*x*(1-x). The fixed points and tangents are shown.
Also f(f(x)) and f(f(f(f(x))) are also shown. Pull the point M, the maximum of the map, up and down to experiment with this.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)
3. Renomalization
This worksheet illustrates the process of period doubling in a slightly different way.
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)

## Tools

Tools for the pre-release version of GeoGebra are here.

This work is licenced under a Creative Commons Licence.