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Math Help - Prove similar triangles in parallelogram

  1. #1
    Junior Member BERMES39's Avatar
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    Prove similar triangles in parallelogram

    Given a parallelogram ABCD and 2 segments FH and EG intersecting the diagonal AC at point P and terminating on 2 opposite parallelogram sides respectively, plus segments HG and EF.

    Pprove that triangles EFP and HGP are similar.

    Could I get a hint on how to solve this problem?

    If it could be proven that HG and EF are parallel or by the same token, that angle AFE=GHC, the proof would follow. I have been unable to make any headway.
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  2. #2
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    Re: Prove similar triangles in parallelogram

    first, show \Delta HCG ~ \Delta FAE using angle-angle.

    showing these two triangles are similar will show m\angle{AFE} = m\angle{CHG}

    m\angle{CHP} = m\angle{AFP} since they are alternate-interior angles

    ... now you can show m\angle{GHP} = m\angle{EFP}. All you need is one more pair of angles (which should be obvious from the diagram)
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  3. #3
    Junior Member BERMES39's Avatar
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    Re: Prove similar triangles in parallelogram

    Thank you skeeter. I still don't see how triangles HCG and FAE are proven similar. Angle EAF=HCG. But to say that angle AEF=CFH or AFE=CHG, would assume that EF is parallel to HG which I still don't see how to prove.
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  4. #4
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    Re: Prove similar triangles in parallelogram

    ... you are correct. I'll have to stare at it awhile and see if I can draw a conclusion.

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    Junior Member BERMES39's Avatar
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    Re: Prove similar triangles in parallelogram

    Thanks. I have been at it several days. I drew it in a computer and the lines (EF and FG) are indeed parallel, just can't figure out how to prove it.
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    Re: Prove similar triangles in parallelogram

    Triangles AEP and CGP are similar so mAP = PC, mEP = GP and mAE = CG for some constant m.
    Triangles APF and CPH are similar so nAP = PC, nAF = CH and nFP = HP for some constant n.
    Equating PC from both sets of equations shows that m = n, in which case, (replacing n by m), mEP = GP and mFP = HP.
    Now feed that information into the triangles EPF and GPH together with the common angles EPF and GPH and we can show that the two triangles are similar. (There might be some well known theorem that, for the moment, I've forgotten, but failing that use of the cosine rule in the two triangles shows that mEF = HG).
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  7. #7
    Junior Member BERMES39's Avatar
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    Re: Prove similar triangles in parallelogram

    Thank you very much, BobP
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    Re: Prove similar triangles in parallelogram

    Quote Originally Posted by BERMES39 View Post
    Thank you very much, BobP
    one simple construction consisting of two lines makes the solution easy
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  9. #9
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    Re: Prove similar triangles in parallelogram

    I was wrong in my assumption about construction lines.Using Bob's similar triangles
    Triangles AEP-CGP AP/PC =PF/HP
    Triangles APF-CPH AP/PC= EP/PG
    PF/HP=EF/PG
    mEPF =mHPG
    Triangles EPF -HPG
    SAS theorem
    If two sides and the included angle of one are in same proportion as the correspondingsides and included angle ofother they are similar
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