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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

Join $AB$. Label $\angle CAB$ as $\alpha$, then find angles $EAB$, $CDB$ and $EFB$ all in terms of $\alpha$. You will need to use the fact that the opposite angles of a cyclic quadrilateral add up to $180$ degrees.

If you are not sure why then see the article on Cyclic Quadrilaterals.

What did you notice about the line segments $CD$ and $EF$ as $C$ and $D$ move around thecircle ?

By considering two of these angles can you now prove what your eyes told you about $CD$ and $EF$?

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