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Thread: Planar Geometry

  1. #1
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    Planar Geometry

    Problem: Given a triangle ABC, be X the point that divides (A,B) in a way that AX=2XB and Y the point that divides (B,C) in a way that BY=3YC. Prove that the lines CX and AY are concurrent and express the concurrent point P in terms of A, AB, AC.

    Answer: P=A+2AB/9+2AC/3

    Commentary: I'm pulling my hair about this problem. I've already proved that CX and AY are concurrent, but I can't see a way to express P in terms of A, AB, AC. If a kind heart could shed some light in this problem, I would be very grateful. Sorry my English.
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  2. #2
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    Re: Planar Geometry

    You just need a little analytic geometry to follow the following:

    Thanks from MacstersUndead
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  3. #3
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    Re: Planar Geometry

    Well, looking at the above answer, I finally saw the way to reach the answer, but I did in a somewhat "easier" way.

    First: Proving that the lines AY and CX are concurrent
    So, if lines AY and CX are non-concurrent, i.e. parallel, then there are a real number α that satisfies the condition AY=α[B]CX[/B]. Expressing AY and CX in terms of AB, AC, we have:

    AY=AB+BY

    To find BY we do:

    AC=AB+BY+YC
    AC=AB+BY+BY/3
    BY=3(AC-AB)/4

    So AY will be:

    AY=AB+3(AC-AB)/4
    AY=(AB+3AC)/4


    CX=CA+AX

    To find AX we do:

    AB=AX+XB
    AB=AX+AX/2
    AX=2AB/3

    So CX will be:

    CX=2AB/3-AC

    That way we can find α:

    AYCX
    (AB+3AC)/4=α(2AB/3-AC)
    (AB+3AC)/4=αAB/4+3αAC/4

    If AY and CX were parallel, then this would mean that 8/3=α=-4/3, what is impossible. Ergo, the lines AY and CX are concurrent.

    Second: Express the concurrent point P in terms of A, AB, AC
    If the points A, P, Y are collinear, there is λ that satisfies APAY, and if the points C, P, X are collinear, there is μ that satisfies CPCX. So, for P we have:

    P=A+AP=A+λAY
    P=C+CP=C+μCX

    A+λAY=C+μCX
    A+λAY=(A+AC)+μCX
    λAY=ACCX

    Using the values previously found for AY and CX, we have:

    (λ/4)AB+(3λ/4)AC=(2μ/3)AB+(1-μ)AC

    Which provides the system λ/4=2μ/3 and 3λ/4=1-μ. The solution to this system is λ=8/9, μ=1/3. So, going back to P:

    P=A+λAY=A+(8/9)AY=A+(8/9)(AB+3AC)/4

    P=A+2AB/9+2AC/3
    Thanks from MacstersUndead
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