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 Δ, the following inequality holds:

a2+b2+c243Δ.

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:

Δ=(a+b+c)(a+bc)(b+ca)(c+ab)4=142(a2b2+a2c2+b2c2)(a4+b4+c4).

First method

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

(a2b2)2+(b2c2)2+(c2a2)202(a4+b4+c4)2(a2b2+a2c2+b2c2)04(a4+b4+c4)34(a2b2+a2c2+b2c2)3(a4+b4+c4)+2(a2b2+a2c2+b2c2)32(a2b2+a2c2+b2c2)(a4+b4+c4)(a2+b2+c2)23(4Δ)2,

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.

a2+b2+c2ab+bc+ca3(a2+b2+c2)(a+b+c)2a2+b2+c23(a+b+c)(a+b+c3)3a2+b2+c23(a+b+c)(a+b+c)(ab+c)(a+bc)a2+b2+c243Δ.

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:

16(a2+b2+c243Δ)

and therefore greater than or equal to 0.

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