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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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OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Cyclic Triangles

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

Think about what stays the same and what changes if you fix $c$ and vary $a+b$.

For the first part you have a right-angled triangle so you might try to use Pythagoras' Theorem which suggests working with $(a+b)^2$. The key idea is to maximise the area of the triangle.

To generalise this go from Pythagoras' Theorem to the Cosine Rule. Again use the area of the triangle. The key idea here is that if you fix $c$ and vary $a+b$ the angle $\angle ACB$ is constant.

The last part calls for a careful argument based on four applications of the second result.