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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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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?

Right Angles

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2
The problem Triangles in Circles will help if you are having difficulty calculating angles.

Try an even number of points round the edge.

Proving this will be easier if you join all the points to the centre and look for isosceles triangles.