Circles in a pendant in a star
- Four vertically-aligned touching circles of equal radius are inscribed in a pendant (the black quadrilateral) inside a regular star as shown in the figure below. Show that the centre of the green circle (the second circle from the top) is on the horizontal segment of the star.
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In fact, say there are 2n such circles (in the above question, n=2) inscribed in the pendant. Show that the n-th circle from the top has its centre on the horizontal segment of the star.
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On the other hand, when there are 2n+1 circles, show that the n-th circle from the top is tangent to the horizontal segment of the star.
I can assure you that these are not independent puzzles and can all be answered in one shot.
Inspired by this wonderful puzzle by Dan.