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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.

Two Regular Polygons

Age 14 to 16 Challenge Level:

This problem draws on angle properties of polygons and factors of 360. Students need to work systematically to find solutions and then reason carefully to justify the completeness of their method.

There is great value in comparing the combinations that work for 81, for 27 and for 54, and then accounting for any observed connections between the relative sizes of angles and number of sides.

Some rich directions in which to open up the task might include:
Is there an infinite number of combinations that will make 81, or for that matter, any specified angle? If not, can we know how many there will be?

It is to be hoped that this kind of questioning allows students to reflect on the extent to which the unit of angle is arbitary (the degree as one part in 360 for a complete rotation). Although 81, or another number, has no decimal part, the two angles that together make that sum may have decimal parts, or at least that possibility needs considering . . .