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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 11 - More on Circles

Age 11 to 16 Challenge Level:

A reminder:

$C = \pi \times d$ or $C = 2 \times\pi \times r$

Where C is the circumference of a circle and $\pi$(pi) is equal to 3.14159...

In terms of LOGO it means that circles of any diameter, radius or circumference can be drawn. Consider the following procedure:

TO CIRC :C
REPEAT 360 [ FD :C/360 RT 1]
END

What do you think this is about?
Once decided, trace the procedure through in your mind's eye.
If you can, talk to others about what you think is happening. If in doubt out check your thoughts by typing in the procedure and testing what it does. N.B. Pi is a primitive approximately equal to 3.14159.

Try CIRC 314

For now experiment by changing:
The number of times you repeat the instruction (360) Or the length of the circumference (:C) Or the amount of turn done after each forward movement (1 degree)

Alternatively you might like to consider the next procedure:

TO CIR :D
REPEAT 360 [FD :D*PI/360 RT 1]
END
Try the following CIR 100

What do you notice now?