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Math Help - Bisecting Diagonals of Quadrilateral means Parallelogram Proof

  1. #1
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    Cool Bisecting Diagonals of Quadrilateral means Parallelogram Proof

    I am trying to prove: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram, by using vectors.

    This is what I have:
    Let ABCD be a quadrilateral with diagonals AC, BD bisecting each other.
    Then DC =1/2 (AC - BD),
    and AB = 1/2 (DB - CA).

    How do I equate these? Am I allowed to say -AB = BA = 1/2 (AC - BD)?
    Thanks for your help.
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  2. #2
    Moo
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    Hello,

    How do I equate these? Am I allowed to say -AB = BA = 1/2 (AC - BD)?
    Yes you can ^^
    The line I'm going to write is not very formal, but it helps you memorize..
    Inverting the two points of a vector will make its direction different, so it will be the opposit of the initial vector

    \vec{MN}=-\vec{NM}

    This goes for any points M and N
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  3. #3
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    Hello, Ultros88!

    If the diagonals of a quadrilateral bisect each other,
    then the quadrilateral is a parallelogram. (vector proof)
    Code:
                A                 B
                * - - - - - - - - *
               /  *           *  /
              /     *     *     /
             /        *        /
            /     *   E *     /
           /  *           *  /
          * - - - - - - - - *
          D                 C

    We are given: . \begin{array}{cccc}\overrightarrow{AE} &=& \overrightarrow{EC} & {\color{blue}[1]}\\ \overrightarrow{EB} &=& \overrightarrow{DE} & {\color{blue}[2]}\end{array}


    We see that: . \begin{array}{cccc}\overrightarrow{AB} &=& \overrightarrow{AE} + \overrightarrow{EB} & {\color{blue}[3]}\\ \overrightarrow{DC} &=& \overrightarrow{DE} + \overrightarrow{EC} & {\color{blue}[4]} \end{array}

    Substitute [1] and [2] into [3]: . \overrightarrow{AB} \:=\:\overrightarrow{EC} + \overrightarrow{DE}

    . . Hence: . \overrightarrow{AB} \:=\:\overrightarrow{DC}


    Since \overrightarrow{AB} \parallel \overrightarrow{DC}\text{ and }|\overrightarrow{AB}| = |\overrightarrow{DC}|, then ABCD is a parallelogram.


    Theorem: If two sides of a quadrilateral are parallel and equal,
    . . . . . . . the quadrilateral is a parallelogram.
    .
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