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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?

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Compare Areas

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

Circle Scaling

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

Circle being scaled

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.