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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?

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Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

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Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Spokes

Age 16 to 18 Challenge Level:

circle and 3 sticks Draw three equal line segments in a unit circle so that they divide the disc into four parts of equal area.

What length do the line segments have to be?

Generalise this result.