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Symmetric Trace

Points off a rolling wheel make traces. What makes those traces have symmetry?

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Mapping the Wandering Circle

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

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Like a Circle in a Spiral

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

The Line and Its Strange Pair

Age 14 to 16 Challenge Level:

$a$ is the point on the line such that $Oa$ is perpendicular to the line, and A is its pair. For any point p on the line, it's pair $P$ forms a trianle $OAP$ which is similar to triangle $Oap$, since $$\frac{|OP|}{|OA|} = \frac{1/|Op|}{1/|Oa|} = \frac{|Oa|}{|Op|}$$. Therefore $\angle OPA$ is a right angle, and as p moves along the line, P traces out a cirlce with diameter $OA$.