Geometric Inequality: Prove √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z)

In summary, the conversation is discussing how to prove the inequality √(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c, using the Ravi-transformation and known inequalities such as AM-GM and the rearrangement inequality. The suggestion to try squaring the expressions is given, specifically squaring the alternative expression of the inequality.
  • #1
dengulakungen
3
0

Homework Statement


Let a,b and c be lengths of sides in a triangle, show that
√(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c

The Attempt at a Solution


With Ravi-transformation the expressions can be written as

√(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z).

Im stuck with this inequality. Can´t find a way to use any known inequalities such as AM-GM or the rearrangement inequality.
 
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  • #3
Yes I have tried to square the experisions, but without success. I will try it again.
 
  • #4
dengulakungen said:
Yes I have tried to square the experisions, but without success. I will try it again.

Your alternative expression of the inequality is the way to go. Even simpler: verify that for any two numbers ##x,y>0## we have ##\sqrt{x+y} \geq \frac{1}{2} \sqrt{2x} + \frac{1}{2} \sqrt{2y}##. Again, have you tried squaring?
 
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What is geometric inequality?

Geometric inequality is a mathematical concept that states that in a given set of geometric figures, certain relationships or inequalities between their measurements or properties always hold true.

What is the significance of the inequality √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z)?

This inequality is important because it provides a way to compare the sums of square roots of different values, and can be used to prove other mathematical theorems.

What does the notation √(x) mean?

The notation √(x) represents the square root of a number, or the value that, when multiplied by itself, gives the original number.

How can this inequality be proven?

This inequality can be proven using various mathematical techniques, such as algebraic manipulation, geometric proofs, or mathematical induction.

What are some real-world applications of geometric inequality?

Geometric inequality can be applied in fields such as physics, engineering, and economics to analyze and optimize various systems and structures.

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