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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?

Compare Areas

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Let AB = BC = 10cm
Have a go at drawing each diagram.

Of the two squares, which has the greatest area?
How do you know?

Draw the radii at the points where the circle touches the triangle.
If the radius of the circle is r, can you find the hypotenuse of the triangle in terms of r?