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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.

LOGO Challenge 10 - Circles

Age 11 to 16 Challenge Level:

It is not easy to create concentric circles and quite alot of work is needed in calculating the diameters of circles based on the length of the side of your many sided polygon.

A step size of 1 unit and a turn of 1 degree produces a circle of circumference 360 units.
A step size of 2 units and a turn of 1 degree produces a circle with a circumference of 720 units.

From these you can work out the diameter as

$C = \pi d$.

That should help you work out where you need to start the next circle.
Some LOGO programs, like MSWLogo understand "PI" and will do calculations which include it. For example: PI * 2 will give a decimal answer.