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Math Help - a:b::b:c when?

  1. #1
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    a:b::b:c when?

    Oh my gosh I just realized that the famous ratio "a is to b as b is to c" is not always true.

    This is quite a shocker for me.

    Can someone explain what conditions the scalene triangle has to fulfill for this rule to hold true? Is it only true for iscoceles?


    Thanks
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  2. #2
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    Hello, rainer!

    Oh my gosh.
    I just realized that the famous ratio "a is to b as b is to c" is not always true.
    Why would you think that it's true?
    Of course it's not true for any three numbers!


    This is quite a shocker for me.

    Can someone explain what conditions the scalene triangle has to fulfill
    for this rule to hold true? .Is it only true for iscoceles? .You mean equilateral.

    If a:b \:=\: b:c, then a,b,c form a geometric sequence.

    That is, the three sides are: . a,\;ar,\;ar^2 in increasing order.
    . . WLOG: . r \ge 1.

    Triangle inequality: . \begin{array}{ccccc}<br />
ar^2 + ar & >& a & [1] \\<br />
ar^2 + a &>& ar & [2] \\<br />
ar + a &>& ar^2 & [3] \end{array}


    Since r \ge 1, both [1] and [2] are satsified.

    For [3] to be satisfied: . ar + a \;>\;ar^2

    Divide by a\!:\;\;r + 1 \;>\;r^2 \quad\Rightarrow\quad r^2-r-1\;<\;0 .[1]


    We have an up-opening parabola: . y \:=\:x^2 - x - 1
    . . When is it below the x-axis?
    Answer: between its x-intercepts.

    The x-intercepts are: . x \;=\;\dfrac{1\pm\sqrt{5}}{2}


    Hence, [1] is satisfied when: . \dfrac{1-\sqrt{5}}{2} \:<\:r\:<\:\dfrac{1+\sqrt{5}}{2}

    Since r \ge 1, we have: . 1 \;\le\;r\;<\;\dfrac{1+\sqrt{5}}{2} .(the Golden Mean)


    The triangle must have sides a,\:ar,\:ar^2, where  1 \:\le\:r\:<\:\phi

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  3. #3
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    Thanks. I don't know why but when I learned this ratio I somehow got the impression that it worked for all triangles.
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