Weitzenböck's inequality

In mathematics, Weitzenböck's inequality states that for a triangle of side lengths $a$ , $b$ , $c$ , and area $Delta$ , the following inequality holds:

: $a^2 + b^2 + c^2 geq 4sqrt{3}, Delta.$

Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality.

Proofs

The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron's formula for the area of a triangle:

: $egin{align}Delta & {} = frac{sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b){4} \& {} = frac{1}{4} sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}.end{align}$

First method

This method assumes no knowledge of inequalities except that all squares are nonnegative.

: $egin{align}{} & (a^2 - b^2)^2 + (b^2 - c^2)^2 + (c^2 - a^2)^2 geq 0 \{} iff & 2(a^4+b^4+c^4) - 2(a^2 b^2+a^2c^2+b^2c^2) geq 0 \{} iff & frac{4(a^4+b^4+c^4)}{3} geq frac{4(a^2 b^2+a^2c^2+b^2c^2)}{3} \{} iff & frac{(a^4+b^4+c^4) + 2(a^2 b^2+a^2c^2+b^2c^2)}{3} geq 2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4) \{} iff & frac{(a^2 + b^2 + c^2)^2}{3} geq (4Delta)^2,end{align}$

and the result follows immediately by taking the positive square root of both sides. From the first inequality we can also see that equality occurs only when $a = b = c$ and the triangle is equilateral.

Second method

This proof assumes knowledge of the rearrangement inequality and the arithmetic-geometric mean inequality.

: $egin{align}& & a^2 + b^2 + c^2 & geq & & ab+bc+ca \iff & & 3(a^2 + b^2 + c^2) & geq & & (a + b + c)^2 \iff & & a^2 + b^2 + c^2 & geq & & sqrt{3 (a+b+c)left(frac{a+b+c}{3}
ight)^3} \iff & & a^2 + b^2 + c^2 & geq & & sqrt{3 (a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \iff & & a^2 + b^2 + c^2 & geq & & 4 sqrt3 Delta.end{align}$

As we have used the rearrangement inequality and the arithmetic-geometric mean inequality, equality only occurs when "a" = "b" = "c" and the triangle is equilateral.

External links

*MathWorld | urlname=WeitzenboecksInequality | title=Weitzenböck's Inequality
*" [http://demonstrations.wolfram.com/WeitzenboecksInequality/ Weitzenböck's Inequality] ," an interactive demonstration by Jay Warendorff, The Wolfram Demonstrations Project.

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