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    #1
    Let points $A$ and $B$ be given on the plane. The mid point of $A$ and $B$, call it $M$, is also given. Mark an arbitrary point $P$ on the plane. Using unmarked straight edge only, construct the line passing through $P$ and parallel to $AB$.

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    #2
    Quote Originally Posted by caffeinemachine View Post
    Let points $A$ and $B$ be given on the plane. The mid point of $A$ and $B$, call it $M$, is also given. Mark an arbitrary point $P$ on the plane. Using unmarked straight edge only, construct the line passing through $P$ and parallel to $AB$.


    Draw a random line through $M$, meeting $AP$ at $X$, and $BP$ at $Y$. Let $Z$ be the point of intersection of $BX$ and $AY$. I'll leave you to figure out why $PZ$ is parallel to $AB$.

    Hint: What this does is to construct the line connecting $P$ to the of $M$ on the line $AB$ (which happens to be the point at infinity).
    Last edited by Opalg; October 15th, 2012 at 14:24.

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    #3 Thread Author
    Quote Originally Posted by Opalg View Post
    Draw a random line through $M$, meeting $AP$ at $X$, and $BP$ at $Y$. Let $Z$ be the point of intersection of $BX$ and $AY$. I'll leave you to figure out why $PZ$ is parallel to $AB$.

    Hint: What this does is to construct the line connecting $P$ to the of $M$ on the line $AB$ (which happens to be the point at infinity).
    Thank You. I can conclude now using Ceva's theorem.

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