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Thread: Help with a proof

  1. #1
    Newbie
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    Oct 2008
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    Help with a proof

    Could I please get some help with the following?
    Given a quadrilateral with points labeled sequentially A B C D with a line segment AC and a line sement BD in the interior noted at intersection by point E with the following givens: segment BC = segment CD and segment AB = segment AD


    Prove that AC is perpendicular to BD I'm stuck!
    thanks SO much for any help or hints
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  2. #2
    Super Member

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    Hello, murphmath!

    Given a quadrilateral $\displaystyle ABCD$ with diagonals $\displaystyle AC$ and $\displaystyle BD$ intersecting at point $\displaystyle E$,
    with the following givens: $\displaystyle BC = CD$ and $\displaystyle AB = AD.$

    Prove that $\displaystyle AC$ is perpendicular to $\displaystyle BD.$
    We have a kite-shaped figure . . .
    Code:
                A
                *
              * | *
            *   |   *
        D * - - + - - * B
           *    |E   *
            *   |   *
             *  |  *
              * | *
               *|*
                *
                C

    We are given: .$\displaystyle AB = AD,\;BC = CD$

    Point $\displaystyle A$ is equidistant from $\displaystyle B$ and $\displaystyle D.$
    Point $\displaystyle C$ is equidistant from $\displaystyle B$ and $\displaystyle D.$

    Hence, $\displaystyle AC$ is the perpendicular bisector of $\displaystyle BD.$

    . . Therefore: .$\displaystyle AC \perp BD$

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