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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

Two circles intersect at $A$ and $B$. $C$ and $D$ are points on one circle and they can be moved around the circle. The line $CA$ meets the second circle in $E$. The line $DB$ meets the second circle in $F$.

As $C$ and $D$ move around one circle what do you notice about the line segments $CD$ and $EF$?

Prove your assertion.


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