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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.

Semi-detached

Age 14 to 16 Challenge Level:

Why do this problem :

This problem could work well as a 'poster' - a visual challenge placed where students will see it. Or presented at the end of a lesson as something to try to solve.

Possible approach :


This printable worksheet may be useful: Semi-detached.

This problem can worked well as something short and closed, but there is also an opportunity to invite questions which open up beyond the initial challenge. Finding the area of a square in a quadrant or squares fitted in the space between either of the two squares in the main problem and the circle.

Key questions :

  • What is the challenge and how might you start ?
  • How did you do it ? Can you explain ?
  • How might you extend this problem ?
  • Can you calculate the area of squares fitted into other places within this diagram ?

Possible extension :

The approach suggested above indicates one route to extension within this context, or for another challenge fitting squares into shapes try Squirty. The problem Semi-square offers another opportunity to work out areas of squares inside circles.

Possible support :

Tilted Squares could be an excellent and accessible challenge for slightly less experienced students.