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Using Geogebra to Investigate Angle Theorems Follow these instructions to create a Geogebra file that explores the relationship between the angle at the centre and the angle at the circumference of a circle.

Open Geogebra Use the View menu to remove the axes and the grid Close the left hand algebra box.

Use the Option menu to ensure Labelling is set to No New Objects

Use this tool to create a circle. The first click on the drawing pad defines the centre, the second a point on the circumference.

Use this tool to place three new points on the centre of the circle.

Now switch to the Arrow tool.

Place the mouse over the centre of the circle and right click.

Rename: name the centre O Similarly rename and label the three points on the circumference ABC in clockwise order.

Use this tool to create 4 line segments: AO, CO, AB and CB.

Use this tool to measure angle AOC and angle ABC. Select these points in this order. What happens if you use a different order?

Now use the Arrow key to select any of points A,B or C and drag them around the circle.

Students can explore this and should be able to conclude that the angle at the centre is always double the angle at the circumference ( allowing for rounding errors).

It is straightforward to draw similar diagrams to illustrate other circle properties.

Exploration of the other tools available should show you how to create similar diagrams to illustrate properties of angles on parallel lines or in polygons.

This tool measures the distance between two points and can be used to demonstrate that the length of two tangents from a point to a circle are equal.

It is possible to place a text box in the diagram which demonstrates, for example, that opposite angles of a cyclic quadrilateral are equal. The syntax for this is Latex and guidance on this syntax is beyond the scope of this document.

Dynamic Web Pages Please refer to the “Exploring Functions in Geogebra� document for information on how to create dynamic web pages, which may be easier for students to use in class than Geogebra files. Editing these in Word will enable teachers to give hints about the expected conclusion without the complexities of using Latex formulae.

Neil Marshall October 2008.