Flower
The proof depends on identifying two sets of similar triangles and spotting that they are arranged around the centre of the inner circle in such a way that they can be used to show that they fit together exactly as the angles add up to 360 degrees. The method is elementary but calls for a systematic approach.
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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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A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
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?