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Math Help - [SOLVED] Concurrency

  1. #1
    RE1
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    Question [SOLVED] Concurrency

    Can someone help me with that proof.
    In triangle IEG
    H is on GI such that EG+GH=EI+IH
    J is on EI such that GI+IJ=EG+EJ
    F is on EG such that GI+GF=EI+EF
    Prove that EH,GJ,IF are concurrent
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let EF=a, \ FG=b, \ GH=c, \ HI=d, \ IJ=e, \ JE=f

    We have:

    \left\{\begin{array}{ll}e+f+a=d+c+b\\a+b+f=c+d+e\e  nd{array}\right.\Rightarrow b=e

    \left\{\begin{array}{ll}c+d+e=a+b+f\\d+e+f=a+b+c\e  nd{array}\right.\Rightarrow c=f

    \left\{\begin{array}{ll}a+e+f=b+c+d\\d+e+f=a+b+c\e  nd{array}\right.\Rightarrow a=d

    Then, \frac{FE}{FG}\cdot\frac{HG}{HI}\cdot\frac{JI}{IE}=  \frac{a}{b}\cdot\frac{c}{d}\cdot\frac{e}{f}=1

    And, from Ceva's theorem, EH, GJ, IF are concurrent.
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