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Thread: Parallelogram

  1. #1
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    Parallelogram

    A Parallelogram $\displaystyle ABCD$ is given. You draw a line which is parallel to $\displaystyle AB$ and crosses the sides [tex]AD[math and $\displaystyle BC$ in $\displaystyle S$ and $\displaystyle Q$.
    Another line which is parallel to $\displaystyle BC$ crosses the sides$\displaystyle AB$ and $\displaystyle DC$ in$\displaystyle P $and $\displaystyle R$. Show that the line that goes through $\displaystyle SR$ and the line that goes through $\displaystyle PQ$ intersect with the line that goes through $\displaystyle AC$ in point $\displaystyle T$. When does that happen & when are those 3 lines parallel..
    (see attachment).

    -------------

    Here's how far i've managed to get..

    $\displaystyle \vec{AB} = \vec{a}$
    $\displaystyle \vec{AD} = \vec{BC} = \vec{b}$

    $\displaystyle \vec{AP} = \alpha\vec{AB} = \alpha\vec{a}$
    $\displaystyle \vec{BQ} = \vec{AS} = \beta\vec{AD} = \beta\vec{b}$
    $\displaystyle \vec{AT} = \gamma\vec{AC} = \vec{AP} + \delta\vec{PQ} $

    $\displaystyle \vec{PQ} = \vec{PB} + \vec{BQ} = (1 - \alpha)\vec{a} + \beta\vec{b} \Rightarrow$
    $\displaystyle \Rightarrow \vec{AT} = \gamma(\vec{a} + \vec{b}) = \alpha\vec{a} + \delta((1 - \alpha)\vec{a} + \beta\vec{b})$

    From that we can see:

    $\displaystyle \vec{a} (\gamma - \alpha - \delta + \delta\alpha) + \vec{b}(\gamma - \delta\beta) = 0$

    $\displaystyle \gamma - \alpha - \delta + \delta\alpha = 0$
    $\displaystyle \gamma - \delta\beta = 0 \Rightarrow \gamma = \delta\beta$

    $\displaystyle \delta\beta - \alpha - \delta + \delta\alpha = 0$
    $\displaystyle \delta(\beta - 1 + alpha) = 0$
    $\displaystyle \delta = \frac{\alpha}{\beta - 1 + \alpha}$

    Now we know when the lines are parallel.. If we get 0 in the denominator then $\displaystyle \delta$ will have an indefinite value -> meaning the lines will never intersect..

    $\displaystyle \vec{AC} \parallel \vec{PQ} \Longleftrightarrow \alpha + \beta = 1$

    ater that i'm lost.. what do i need to do now and how?
    Attached Thumbnails Attached Thumbnails Parallelogram-parallelogram.jpg  
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