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# Circle Scaling

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### Triangle Midpoints

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### The Medieval Octagon

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Age 14 to 16

Challenge Level

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.

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.