Chris Sangwin's GeoGebra page
The following pages contain interesting geometry illustrated with GeoGebra. GeoGebra is a Java applet,
freely available (licence: GPL) from its homepage.
Classical Geometry
- A circle theorem
This worksheet allows you to experiment with one classical circle theorem.
- Trisect an angle I
One attempt to divide an arbitrary angle into three equal parts, which does not work!
- Trisect an angle II
Another attempt to divide an arbitrary angle into three equal parts.
- Pascal's Theorem
Take six points on a conic section. The intersection of lines joining them are colinear.
- Tangents to a conic section
Using Pascal's Theorem we can construct the tangent to a conic through a point on the conic.
- Morley's Theorem
Inside every triangle is an equilateral triangle trying to escape!
- The nine point circle
This construction shows the "nine point" circle and Euler line.
- Tangents to two circles
This construction shows how to construct lines tangent to two circles.
Problems
- Concurent circles I
Why are these three circles concurrent?
- Concurent circles II
Why are these four circles concurrent?
- Touching circles
How do you construct three circles which touch?
- Circle, segment and lines
A segment moves around a circle......
Complex numbers
- Squaring complex numbers
This demonstrates what happens to lines and circles when points are multiplied by themselves.
- 1/z
This demonstrates what happens to lines and circles when we take the reciprocal of points.
Mobius Transformation
This demonstrates the Mobius transformation.
Mechanisms
- Watt's linkage
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
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Cobwebbing
This illustrates the procedure of "cobwebbing" - a graphical method of looking at the long term behaviour of one dimensional maps.
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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.
- Renomalization
This worksheet illustrates the process of period doubling in a slightly different way.
Tools
Tools for the pre-release version of GeoGebra are here.
This work is licenced under a Creative Commons Licence.