Lunar Angles
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.
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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?
Kissing
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?
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.