Weitzenböck's inequality

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

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

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

1 Proofs
- 1.1 First method
- 1.2 Second method
2 External links

[edit] 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:

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

[edit] First method

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

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

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.

[edit] Second method

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

	$a^2 + b^2 + c^2 \,\;$	$\geq ab+bc+ca \,\;$
$\iff$	$3(a^2 + b^2 + c^2) \,\;$	$\geq (a + b + c)^2 \,\;$
$\iff$	$a^2 + b^2 + c^2 \,\;$	$\geq \sqrt{3 (a+b+c)(\textstyle{\frac{a+b+c}{3}})^3} \,\;$
$\iff$	$a^2 + b^2 + c^2 \,\;$	$\geq \sqrt{3 (a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \,\;$
$\iff$	$a^2 + b^2 + c^2 \,\;$	$\geq 4 \sqrt3 \Delta \,\;$

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.