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Math Help - A nice, unsolved geometry problem

  1. #1
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    A nice, unsolved geometry problem

    The following geometry question looks simple but still unsolved.
    Fox 223 (I am walking 8 foxes at the top of my head and trying to keep their tails apart !)
    Creative Unusual Geometry Problems: Fox 223

    This also looks related to 223:
    Creative Unusual Geometry Problems: Fox 224
    Attached Thumbnails Attached Thumbnails A nice, unsolved geometry problem-8foxes.com_223.gif  
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  2. #2
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    There is a theorem, (easily proved using similar triangles), involving the intersection of two chords in a circle.
    If as a result of the intersection, one chord splits into lengths a and b, while the other splits into lengths p and q, then ab = pq.
    Apply this at each of the four corners of the rhombus, add the four equations, cancel this down and you get the required equation.
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  3. #3
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    I think the theorem which states: "ab = pq" must be Ptolemy. But I am not sure about its name.
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  4. #4
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    No, this is not Ptolemy's theorem.
    For this one, put two intersecting chords in a circle. Call them say AB and PQ and call the point of intersection C.
    By the angles on the same arc being equal theorem you have a choice of two sets of similar triangles, ACQ and PCB, or ACP and BCQ.
    Choose one of the two, equate corresponding ratios, cross multiply and you have your result.
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