PlanetMath: Weizenbock's inequality

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Weizenbock's inequality

(Theorem)

In a triangle with sides , , , and with area , the following inequality holds:

$\displaystyle a^2 + b^2 + c^2 \geq 4A \sqrt{3}$

The proof goes like this: if $s=\frac{a+b+c}{2}$ is the semiperimeter of the triangle, then from Heron's formula we have:

$\displaystyle A = \sqrt{s(s-a)(s-b)(s-c)}$

But by squaring the latter and expanding the parentheses we obtain:

$\displaystyle 16A^2 = 2(a^2 b^2 + a^2 c^2 + b^2 c^2) - (a^4 + b^4 + c^4)$

Thus, we only have to prove that:

$\displaystyle (a^2 + b^2 + c^2)^2 \geq 3[2(a^2 b^2 + a^2 c^2 + b^2 c^2) - (a^4 + b^4 + c^4)]$

or equivalently:

$\displaystyle 4(a^4 + b^4 + c^4) \geq 4(a^2 b^2 + a^2 c^2 + b^2 c^2)$

which is trivially equivalent to:

$\displaystyle (a^2 - b^2)^2 + (a^2 - c^2)^2 + (b^2 - c^2)^2 \geq 0$

Equality is achieved if and only if

(i.e. when the triangle is equilateral) .

See also the Hadwiger-Finsler inequality, from which this result follows as a corollary.

"Weizenbock's inequality" is owned by mathcam. [ full author list (2) | owner history (1) ]

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