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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

There is something special about a triangle with sides of $8, 15$ and $17$.
The sides of the triangle are tangents to the circle.
You might like to look at how to construct an inscribed circle in the Thesaurus (Search under constructions).
The angle between a tangent and a radius is $90^\circ$ .
The centre of the inscribed circle lies on the bisector of the angle between the two tangents from any point outside the circle.
The two tangents to a circle from the same point are equal.