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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


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?


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:

REPEAT :Y [FD :X LT 360/:Y]

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

What happens as the value for :Y increases yet further? (i.e. considering the limitations with your monitor's powers of resolution.)
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.