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Math Help - Apollonius Theorem

  1. #1
    Member jacs's Avatar
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    Apollonius Theorem

    Hi, i am having trouble with this proof. I have found mulitple webpages on the theorem, but not one of them i coudl find offers me a 'how to'

    It is to do with Appolonis' Theorem. The question is:

    "In triangle ABC, D is the midpoint of BC, then prove that -
    AB² + AC² = 2AD² + 2CD²

    Hint: Use Pythagoras' Theorem."


    although from looking at the web, i am thinking my teacher has copied it down incorrectly since they all seem to say AB² + AC² = 2AD² + (BC²)/2


    since AD is a median, and it is not known that there are any right angles in this triangle, i am not sure how to get Pythagoras in there at all

    thanks for any help

    jacs
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  2. #2
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    Krizalid's Avatar
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    You can use Stewart's theorem.
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  3. #3
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    Hello, jacs!

    In \Delta ABC,\:D is the midpoint of BC.
    Prove that: . AB^2 + AC^2 \:= \:2\!\cdot\!AD^2 + 2\!\cdot\!CD^2

    We have: . \Delta ABC with median AD.
    Let BD = DC = x.

    Drop perpendicular AE from vertex A to side BC. .Let h = AE.
    .Let a = ED, then: BE = x - a
    Code:
                  A
                  *
                 *:* *
                * : *   *
               *  :  *     *
              *   :   *       *
             *    :h   *         *
            *     :     *           *
           *      :      *             *
        B * - - - + - - - * - - - - - - - * C
             x-a  E   a   D       x
    In right triangle AEB\!:\;\;AB^2\;=\;h^2 + (x-a)^2 . [1]

    In right triangle AEC\!:\;\;AC^2 \;=\;h^2 + (x+a)^2 . [2]

    Add [1] and [2]: . AB^2 + AC^2 \;=\;h^2 + (x-a)^2 + h^2 + (x+a)^2

    And we have: . AB^2 + AC^2 \;=\;2(h^2+a^2) + 2x^2 . [3]

    . . In right triangle AED\!:\;\;h^2 + a^2 \:=\:AD^2
    . . . . We also know that:. x = CD

    Therefore, [3] becomes: . AB^2 + AC^2 \;=\;2\!\cdot\!AD^2 + 2\!\cdot\!CD^2

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  4. #4
    Member jacs's Avatar
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    Red face

    Thanks so much for that. I was of course going about it all wrong and now I have seen that, it is very clear.

    Thanks for a concise explanation too, made it very easy to follow

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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by jacs View Post
    Hi, i am having trouble with this proof. I have found mulitple webpages on the theorem, but not one of them i coudl find offers me a 'how to'

    It is to do with Appolonis' Theorem. The question is:

    "In triangle ABC, D is the midpoint of BC, then prove that -
    AB² + AC² = 2AD² + 2CD²

    Hint: Use Pythagoras' Theorem."


    although from looking at the web, i am thinking my teacher has copied it down incorrectly since they all seem to say AB² + AC² = 2AD² + (BC²)/2


    since AD is a median, and it is not known that there are any right angles in this triangle, i am not sure how to get Pythagoras in there at all

    thanks for any help

    jacs
    As D is the midpoint of BC, 2CD^2 = 2(BC/2)^2=(BC^2)/2.

    RonL
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