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What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

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Spirostars

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Retracircles

Age 16 to 18 Challenge Level:

$BO$ is a tangent to the two equal circles and hence angle $BOC$ is a right angle. If $OC=$3 units then the large outer circle has radius $6$ units. Pythagoras' theorem will give the radius of the circle centre $B$. You cannot assume that angle $BAC$ is a right angle.