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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?

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Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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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?

Lunar Angles

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Although this is an unusual question it is very straightforward and depends only on understanding that the angle between two curves is the same as the angle between their tangents at the point of intersection. This is good preparation for an introduction to non-Euclidean geometry.