MHB Find Min Value: $a,b,c>0$ with $a+b+c=k$

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The discussion centers on finding the minimum value of the expression √(a² + b²) + √(b² + c²) + √(c² + a²) under the constraints that a, b, and c are positive and their sum equals k. Participants share various approaches and solutions to this optimization problem. The focus is on deriving a method to minimize the expression effectively while adhering to the given conditions. The conversation highlights the importance of understanding the relationships between the variables to achieve the minimum value. Overall, the goal is to find a precise mathematical solution for the stated problem.
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$a,b,c>0$

$a+b+c=k$

find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
 
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My Solution:

Given $$a+b+c = k$$ and $$a,b,c>0$$

Now we can write $$\sqrt{a^2+b^2} = \left|a+ib\right|$$ and $$\sqrt{b^2+c^2} = \left|b+ic\right|$$ and $$\sqrt{c^2+a^2} = \left|c+ia\right|$$

Where $$i=\sqrt{-1}$$ So Using Triangle Inequality of Complex number

$$\left|a+ib\right|+\left|b+ic\right|+\left|c+ia\right|\geq \left|\left(a+b+c\right)+i\left(b+c+a\right)\right| = \left|k+ik\right|=\sqrt{2}k$$

and equality hold when $$\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{a}$$
 
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jacks said:
My Solution:

Given $$a+b+c = k$$ and $$a,b,c>0$$

Now we can write $$\sqrt{a^2+b^2} = \left|a+ib\right|$$ and $$\sqrt{b^2+c^2} = \left|b+ic\right|$$ and $$\sqrt{c^2+a^2} = \left|c+ia\right|$$

Where $$i=\sqrt{-1}$$ So Using Triangle Inequality of Complex number

$$\left|a+ib\right|+\left|b+ic\right|+\left|c+ia\right|\geq \left|\left(a+b+c\right)+i\left(b+c+a\right)\right| = \left|k+ik\right|=\sqrt{2}k$$

and equality hold when $$\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{a}$$
nice solution !
 
Albert said:
$a,b,c>0$

$a+b+c=k$

find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
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Thread 'Area of Overlapping Squares'

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