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Thread: Show constructibility

  1. #1
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    Question Show constructibility

    "Constructible lengths are those that can be built from known lengths by the operations of $\displaystyle +$, -, $\displaystyle \times$, $\displaystyle \%$ and $\displaystyle \sqrt$."

    Show, using similar triangles, that if lengths $\displaystyle l_1$ and $\displaystyle l_2$ are constructible, then so are $\displaystyle l_1l_2$ and $\displaystyle l_1/l_2$.
    How do you go about it?




    How do you get a proper minus sign in $\displaystyle LaTeX$? I've tried "-", "$-$", "--", ... nothing works.
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  2. #2
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    Construct a triangle $\displaystyle PQR$ with $\displaystyle PQ$ of length $\displaystyle l_{1}, QR$ of length $\displaystyle 1$ and $\displaystyle RQ$ of any length at all (so long as $\displaystyle PQR$ is a triangle).
    Now construct a line $\displaystyle Q'R'$, parallel to $\displaystyle QR$ and of length $\displaystyle l_{2},$ followed by lines $\displaystyle P'Q'$ and $\displaystyle P'R'$ parallel to there none dash equivalents.
    The two triangles will be similar and if you work out the length of $\displaystyle P'Q'$, it will be seen to be $\displaystyle l_{1}l_{2}.$
    A similar routine can be used to construct a line of length $\displaystyle l_{1}/l_{2}.$
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