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Cyclic quadrilateral-construction

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  • pskssvsk2003
    Perhaps this is obvious to many of you; Can you please suggest how to construct a cyclic quadrilateral given the lengths of its four sides? Is there a solution
    Message 1 of 2 , Feb 2, 2004
      Perhaps this is obvious to many of you;

      Can you please suggest how to construct a cyclic quadrilateral given
      the lengths of its four sides?
      Is there a solution which does not first arithmetically calculate the
      length of its radius using for example, Brahmagupta's formula?

      Sriram Panchapakesan
    • yagcimustafa
      ... given ... the ... This problem is owned by the mathematician Sturm who has borned in Genova in 1803. Suppose the problem has been solved, draw a cyclic
      Message 2 of 2 , Feb 3, 2004
        --- In Hyacinthos@yahoogroups.com, "pskssvsk2003" <pskssvsk2003@y...>
        wrote:
        > Perhaps this is obvious to many of you;
        >
        > Can you please suggest how to construct a cyclic quadrilateral
        given
        > the lengths of its four sides?
        > Is there a solution which does not first arithmetically calculate
        the
        > length of its radius using for example, Brahmagupta's formula?
        >
        > Sriram Panchapakesan

        This problem is owned by the mathematician Sturm who has borned in
        Genova in 1803.
        Suppose the problem has been solved, draw a cyclic quadrilateral ABCD
        and its' circumcircle. We know A+C=B+D=180. Draw the diagonal AC.Then
        take a point on [CD (outside the circle) so that m(BAC)=m(DAE). Since
        m(ABC)=m(ADE), then the triangles ABC and ADE are similar. a/d =
        b/|DE| = |AC|/|AE| so |DE| = bd/a. Since we know |DE|, we also know
        |CE| now. |AC|/|AE|=a/d=constant means that A is a point such that
        the ratio of the distances to C and E is constant. So it must be on
        the Apollonius circle. I think you can easily construct it now.
        Mustafa
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