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Math Help - Find the midpoint of a triangle with three endpoint given

  1. #1
    DBA
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    Find the midpoint of a triangle with three endpoint given

    Hello,
    I try to figure out a formula to find the mid point of a triangle.
    Given are the 3 endpoints A(x,y), B(x,y) and C(x,y).


    I can find the side length a, b,c of the triangle by using the distance formula.

     d = \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2}

    Then I could find the angles with the Law of Cosine.

    A = \cos^{-1}(\frac{b^2+c^2-a^2}{2bc})

    B = \cos^{-1}(\frac{a^2+c^2-b^2}{2ac})

    C = \cos^{-1}(\frac{a^2+b^2-c^2}{2ab})


    The intersection, m(x,y), of the angle bisectors is the center of the incircle.
    But how can I find this intersection with the information I have?

    Thanks for any help!
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  2. #2
    MHF Contributor

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    If we can use vectors then this is useful.
    Let p = \overrightarrow {AB} \;,\,q = \overrightarrow {AC} then the vector r=\left\| p \right\|q + \left\| q \right\|p bisects the angle \angle BAC.
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  3. #3
    DBA
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    Hello,
    thanks for the answer but I do not understand it.
    I am searching for the midpoint m(x,y).
    I did not work with vectors yet. So, what exactly does p and q stands for? And how do I get the x-value and the y-value of the point?
    I cannot find the mid point by drawing it because I want to compute it by a computer.

    Thank you.
    Last edited by DBA; November 26th 2010 at 01:48 PM. Reason: typo
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  4. #4
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    Clearly if you do not understand vectors, then that approach is useless to you.
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  5. #5
    DBA
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    Ok, thank you. I will try to read myself into vectors to understand it.
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