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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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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?


Stage: 4 Challenge Level: Challenge Level:1

A circle is inscribed in a triangle which has side lengths of $8, 15$ and $17 \; \text{cm}$.

What is the radius of the circle?

Can you adapt your method to find the radius of a circle inscribed in any right-angled triangle $ABC$?

Triangle with points A B C