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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

Triangle ABC inscribed in circle with diameter AB
A triangle $ABC$ is inscribed in a circle with $AB$ as diameter. Find the maximum value of $AC + CB$.

Now generalise your result to the case where $AB$ is fixed but not a diameter of the circle.

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius $r$ that has the maximum perimeter and the maximum area.

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