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Fixing It

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Thebault's Theorem

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Created with GeoGebra

Take any parallelogram and draw squares on the sides of the parallelogram. What can you prove about the quadrilateral formed by joining the centres of these squares?

You can change the parallelogram by dragging the vertices.


This dynamic image is drawn using Geogebra, free software and very easy to use. You can download your own copy of Geogebra from together with a good help manual and Quickstart for beginners.