Reeb vector field

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In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths $a$, $b$, $c$, and area $\mathrm{\Delta }$, the following inequality holds:

$a^2+b^2+c^2\ge 4\sqrt{3}\mathrm{\Delta }.$

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:

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

First method

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

$\begin{array}{rl}& (a^2-b^2{)}^2+(b^2-c^2{)}^2+(c^2-a^2{)}^2\ge 0\\ ⟺& 2(a^4+b^4+c^4)-2(a^2{b}^2+a^2{c}^2+b^2{c}^2)\ge 0\\ ⟺& \frac{4(a^4+b^4+c^4)}{3}\ge \frac{4(a^2{b}^2+a^2{c}^2+b^2{c}^2)}{3}\\ ⟺& \frac{(a^4+b^4+c^4)+2(a^2{b}^2+a^2{c}^2+b^2{c}^2)}{3}\ge 2(a^2{b}^2+a^2{c}^2+b^2{c}^2)-(a^4+b^4+c^4)\\ ⟺& \frac{(a^2+b^2+c^2{)}^2}{3}\ge (4\mathrm{\Delta }{)}^2,\end{array}$

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.

$\begin{array}{rlrlrl}& & a^2+b^2+c^2& \ge & & ab+bc+ca\\ ⟺& & 3(a^2+b^2+c^2)& \ge & & (a+b+c{)}^2\\ ⟺& & a^2+b^2+c^2& \ge & & \sqrt{3(a+b+c){\left(\frac{a+b+c}{3}\right)}^3}\\ ⟺& & a^2+b^2+c^2& \ge & & \sqrt{3(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\\ ⟺& & a^2+b^2+c^2& \ge & & 4\sqrt{3}\mathrm{\Delta }.\end{array}$

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.

Third method

It can be shown that the area of the inner Napoleon's triangle is:

$\frac{1}{6}(a^2+b^2+c^2-4\sqrt{3}\mathrm{\Delta })$

and therefore greater than or equal to 0.

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• "Weitzenböck's Inequality," an interactive demonstration by Jay Warendorff - Wolfram Demonstrations Project.