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Pericut

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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Circumspection

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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Quads

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?

Circle-in

Stage: 4 Challenge Level: Challenge Level:1

The thesaurus might be useful to support discussion on the properties of tangents to the circle.

The general rule is elegant.

It might be worth discussing whether this can be adapted for non-right-angled triangles.