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Math Help - Concurrency at interior point

  1. #1
    Senior Member pankaj's Avatar
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    Concurrency at interior point

    Let P be an interior point of \triangle ABC such that the lines AA_{1},BB_{1},CC_{1} are concurrent at P and the points A_{1},B_{1},C_{1} lie on BC,CA,AB respectively.Find using vectors or otherwise value of
    \frac{PA_{1}}{AA_{1}}+\frac{PB_{1}}{BB_{1}}+\frac{  PC_{1}}{CC_{1}}
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let PQ\perp BC, \ AD\perp BC

    Then, \frac{PA_1}{AA_1}=\frac{PQ}{AD}

    A_{\Delta PBC}=\frac{PQ\cdot BC}{2}, \ A_{\Delta ABC}=\frac{AD\cdot BC}{2}

    \frac{A_{\Delta PBC}}{A_{\Delta ABC}}=\frac{\frac{PQ\cdot BC}{2}}{\frac{AD\cdot BC}{2}}=\frac{PQ}{AD}

    So, \frac{PA_1}{AA_1}=\frac{A_{\Delta PBC}}{A_{\Delta ABC}}

    In a similar way we have

    \frac{PB_1}{BB_1}=\frac{A_{\Delta PAC}}{A_{\Delta ABC}} and \frac{PC_1}{CC_1}=\frac{A_{\Delta PAB}}{A_{\Delta ABC}}

    Then \frac{PA_1}{AA_1}+\frac{PB_1}{BB_1}+\frac{PC_1}{CC  _1}=\frac{A_{\Delta PBC}+A_{\Delta PAC}+A_{\Delta PAB}}{A_{\Delta ABC}}=\frac{A_{\Delta ABC}}{A_{\Delta ABC}}=1
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