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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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Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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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.

Circle Packing

Age 14 to 16 Challenge Level:

Equal circles can be arranged in regular square or hexagonal packings to fill space as shown in the diagram so that each circle touches four or six others.

Square and hexagonal packings

What percentage of the plane is covered by circles in each packing pattern? How is this of use in packing cylindrical cans and what are the advantages and disadvantages of the two packing systems?

 

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