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I am trying to solve a geometric problem I came up with for myself, and I've been stuck for hours trying to solve a system of equations. Here it is:

$$(1)\: ab + bc + ac = \frac{1}{2}\:(2)\: \frac{a}{b}=\frac{b}{c}\:(3)\:\frac{b}{c}=\frac{2c}{a}$$

One of the things I arrived at is this elegant but seemingly useless identity:

$$a^2+2b^2+4c^2=1$$

Another one is:

$$a=\frac{1}{2b}-b^2-b$$

I have absolutely no clue how to solve this system or (given I'm on the right path) proceed. Even the slightest hint would be appreciated.

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    $\begingroup$ It would be better if you could provide the original question along with how you have proceeded. $\endgroup$
    – Soham
    Oct 13, 2016 at 17:37

1 Answer 1

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Hint

$$\frac ab=\frac bc \iff ac=b^2$$

$$\frac bc=\frac {2c}{a} \iff ab=2c^2$$

Dividing previous equalities one has

$$\frac{c}{b}=\frac{b^2}{2c^2}\iff b^3=2c^3\iff b=\sqrt[3]{2}c.$$

Now, from $ac=b^2$ you can get $a$ in terms of $c.$

Finally, from $$ab+bc+ac=\frac 12$$ you can get $c.$ And so, $a$ and $b.$

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