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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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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?

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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:

Is it true that p 2 + q 2 = s 2 + r 2

In 2 dimensions, where the rectangle ABCD and the point V are in the same plane?

In 3 dimensions, where the diagram represents a pyramid, with vertex V, on a rectangular base ABCD?