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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?

Circle Scaling

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

If the areas are in the ratio 1:2:3 what can you say about the ratio of the radii?

Can you find some key right-angled triangles that will give you the lengths you are after? Pythagoras' theorem is useful!

Some useful constructions can be found in the Thesaurus if you look up "construction". A perpendicular line might be particularly useful.