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Menelau's theorem

  1. Feb 21, 2008 #1

    disregardthat

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    [SOLVED] Menelau's theorem

    "If a transversal is drawn to cut the sides of a triangle (produced if necessary), the product of the ratios of alternate segments is minus one.

    So, for http://www.freewebs.com/jarle10/menelau's theorem.JPG (See picture)

    [tex]\frac{BU}{UC} \cdot \frac{CY}{YA} \cdot \frac{AZ}{ZB} = -1[/tex]"

    I don't understand this. How can they differ between negative and positive lengths in a plane? In the proof, an obvious step is that AY=-YA. But BZ apparantly equal ZB. This makes it even more confusing. Can someone please explain how and why they differ between negative and positive values of the lengths in the plane?

    I proved the case where the line does not intersect the triangle in a similar way as the first proof. The reason for that the product is minus one beats me. Why differ between positive and negative values?

    Appreciate any help.

    EDIT: I think I got it. When a point is internally dividing a line segment, the ratio is positive, and when it's dividing the segment extrernally, it's negative, as mentioned earlier in the chapter. yay
     
    Last edited: Feb 21, 2008
  2. jcsd
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