Hadwiger-Finsler inequality

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In mathematics, the Hadwiger-Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler. It states that if Δ is a triangle in the plane with side lengths a, b and c and area A, then

$a^{2} + b^{2} + c^{2} \geq (a - b)^{2} + (b - c)^{2} + (c - a)^{2} + 4 \sqrt{3} A \quad \mbox{(HF)}.$

Weitzenböck's inequality is a straightforward corollary of the Hadwiger-Finsler inequality: if Δ is a triangle in the plane with side lengths a, b and c and area A, then

$a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} A \quad \mbox{(W)}.$

Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if Δ is an equilateral triangle, i.e. a = b = c.