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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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Pericut

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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Set Square

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

Flower

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

You need to use the fact that if two circles touch then the line joining their centres goes through the point at which they touch. (Why is that?) Then join the centres of all the 'petals' to the centre of the inner circle and draw the hexagon formed by joining the centres of adjacent 'petals' as in the sketch. After that the proof depends on finding similar triangles.