Question
Let A,B and C be three circles of the same radius, each touching the other two externally.Let D be the small circle enclosed between A,B and C so D touches them all.If r is the radius of the circles A,B and C and s is the radius of the circle D,find the exact value of the fraction (s over r).
Answer
It is very helpful for this type of question to draw a diagram of the circles. To relate the various distances to each other, try the following:
Connect the center points of the 3 large circles to form a triangle. At this point, an important question is whether or not the edges of this triangle go through the points where the large circles touch each other. This is not difficult to answer. Think about how circles fit together in a honeycomb-like pattern. Once you have determined that the triangle goes through the points where the large circles meet, the distances r and s can be related to each other by using the theorem of Pythagoras with suitably chosen triangles. (Hint: You may get two equations with two unknowns, and need to solve a quadratic equation.)
I think that this is a fun problem. See if you can solve it with these hints. If you need more help, feel free to ask again and I will be very happy to provide more information.
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