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Math Help - Paralellogram

  1. #1
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    Paralellogram

    Any cyclic paralellogram having unequal adjacent sides is necessarily a
    (a)Square
    (b)Rectangle
    (c)Rhombus
    (d)Trapezium
    Why?
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  2. #2
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    If a parallelogram is cyclic, then it must be a rectangle. The perpendicular bisectors of the sides of a parallelogram are concurrent only when it is a rectangle. For a parallelogram to be cyclic, its perpendicular bisectors must be concurrent at the center of the circumcircle; therefore, a cyclic parallelogram must be a rectangle.
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  3. #3
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    Hello, devi!

    Any cyclic paralellogram having unequal adjacent sides is necessarily a:
    . . (a) square . (b) rectangle . (c) rhombus . (d) trapezium

    This is my approach to this problem.
    Perhaps you can hammer it into a formal proof.


    Opposite sides of a parallelogram are parallel.
    They are formed by a pair of parallel chords.
    Code:
                  * * *
              *           *
            *               *
         A o o o o o o o o o o B
                      
          *                   *
          *         *         *
        D o o o o o o o o o o o C
                      
           *                 *
            *               *
              *           *
                  * * *

    Opposite sides of a parallelogram are equal.
    They are formed by two chords equidistant from the center.
    Code:
                  * * *
              *           *
          A o o o o o o o o o B
           *        :        *
                    :
          *         :         * C
          *         *         o
          *          \      o *
                      \   o
           *            o    *
            *         o     *
              *     o     *
                  o * *
                D

    Since opposite sides of a parallelogram are parallel and equal,
    the chords are parallel and equidistant from the center.
    Code:
                  * * *
              *           *
          A o o o o o o o o o B
           *        :        *
                    :
          *         :         *
          *         *         *
          *         :         *
                    :
           *        :        *
          D o o o o o o o o o C
              *           *
                  * * *

    Your turn . . .

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