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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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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?


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

The proof depends on identifying two sets of similar triangles and spotting that they are arranged around the centre of the inner circle in such a way that they can be used to show that they fit together exactly as the angles add up to 360 degrees. The method is elementary but calls for a systematic approach.