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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

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Medallions

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

Circle Scaling

Age 14 to 16 Challenge Level:

If the areas are in the ratio 1:2:3 what can you say about the ratio of the radii?

Can you find some key right-angled triangles that will give you the lengths you are after? Pythagoras' theorem is useful!

Some useful constructions can be found in the Thesaurus if you look up "construction". A perpendicular line might be particularly useful.