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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 12 - Concentric Circles

Age 11 to 16 Challenge Level:
What is the relationship between the diameters of the circles?
What about the relationships between the circumferences?

The number of times the turtle turns and the distance it moves after each turn gives you the circumference which means you can calculate the diameter.

Where does the turtle need to be to start to draw each of the circles?
Instead of starting with the circumference - can you draw a circle with a known diameter?