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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

Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?


Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Rectangular Pyramids

Age 14 to 18 Challenge Level:

Why do this problem?

It provides experience of generalising a result from 2 dimensions to an equivalent result in 3 dimensions. This problem asks the question for them but learners should be encouraged to ask themselves "What if..." and always to think about possible generalisations.

Key questions

What comes to mind when a problem involves squares of distances?
If we are looking for Pythagoras theorem where are the right angles triangles?

Possible extension

The problem Pythagoras for a Tetrahedron.