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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:

These challenges arose from conversations and ideas during a Royal Institution Saturday morning Maths Workshop. Consider the procedure:

TO POLY :X :Y
REPEAT :Y [FD :X LT 360/:Y]
END

where :X and :Y are variables, that can be changed as you experiment and try to understand what is happening.

Think about, try to imagine, then experiment with:

POLY 50 7
POLY 20 16
POLY 20 30
etc..

What happens as the value for :Y increases yet further? (i.e. considering the limitations with your monitor's powers of resolution.)
.
.
.
BUT
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What is the most elegant way of coding the drawing of a circle of radius 60 screen units in the centre of your screen?

[Hint: use the primitive HOME to return turtle.]

Naturally, geometric designs have long been embellished with circles.

You might like to consider re-creating the following designs this month.