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- Feb 14, 2012

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- Thread starter anemone
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- Feb 14, 2012

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- Jan 26, 2012

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My solution

$\begin{eqnarray}

a - b \;& \; \le\; & \;c \;& \;\le b-a\\

b - c\; & \;\le\; &\;a &\; \le c-b\\

a - c \;& \;\le\; &\;b & \;\le c-a

\end{eqnarray}.$

From the right hand side of the first inequality we have $a+c \le b$ and the left hand side of the second inequality we have $b \le a + c$ giving the $b=a+c$

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- Feb 14, 2012

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Squaring both sides of the inequality of $|a-b|\ge|c|$, we get

$(a-b)^2\ge c^2$

$(a-b)^2-c^2\ge 0$

$(a-b+c)(a-b-c)\ge 0$---(1)

Similarly we have

$(b-c+a)(b-c-a)\ge 0$

$-(b-c+a)(-b+c+a)\ge 0 $

$\rightarrow -(a+b-c)(a-b+c)\ge 0$---(2)

and $(c-a+b)(c-a-b)\ge 0$

$(-c+a-b)(-c+a+b)\ge 0$

$\rightarrow (a-b-c)(a+b-c)\ge 0$---(3)

Multiply these three inequalities yields

$(a-b+c)(a-b-c)(-(a+b-c)(a-b+c))(a-b-c)(a+b-c) \ge 0$

$-(a-b+c)^2(a-b-c)^2(a+b-c)^2 \ge 0$

Apparently, this inequality holds true if either $a-b+c=0 (a+c=b)$ or $a-b-c=0 (b+c=a)$ or $a+b-c=0 (a+b=c)$.