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Padoa's Inequality

Padoa's inequality states that the side lengths a, b, and c of a nondegenerate triangle satisfy

abc>=(a+b-c)(b+c-a)(c+a-b),
(1)

with equality iff the triangle is an equilateral triangle (Padoa 1925; Bottema et al. 1969, p. 12; Nelsen 2006). If s=(a+b+c)/2 is the semiperimeter, an equivalent form is

abc>=8(s-a)(s-b)(s-c).
(2)

For a proof, put x=s-a, y=s-b, and z=s-c. The triangle inequality gives x,y,z>0, and a=y+z, b=z+x, and c=x+y. Comparing the arithmetic mean and geometric mean of each pair gives

(x+y)(y+z)(z+x)>=2sqrt(xy)2sqrt(yz)2sqrt(zx)=8xyz,
(3)

which proves the result. Equality requires x=y=z, or equivalently a=b=c.

If Delta is the area, R the circumradius, and r the inradius, then Heron's formula, R=abc/(4Delta), and r=Delta/s give

R/r=(abc)/(4(s-a)(s-b)(s-c)).
(4)

Padoa's inequality is therefore equivalent to Euler's inequality R>=2r.

See also

Euler's Inequality, Triangle Inequalities

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References

Alsina, C. and Nelsen, R. B. When Less Is More: Visualizing Basic Inequalities. Washington, DC: Mathematical Association of America, 2009. https://doi.org/10.5948/UPO9781614442028.Bottema, O.; Djordjević, R. Z.; Janić, R. R.; Mitrinović, D. S.; and Vasić, P. M. Geometric Inequalities. Groningen, Netherlands: Wolters-Noordhoff, p. 12, 1969.Nelsen, R. B. "Proof Without Words: Padoa's Inequality (Alessandro Padoa, 1868-1937)." Math. Mag. 79, 53, 2006. https://doi.org/10.1080/0025570X.2006.11953375.Padoa, A. "Una questione di minimo." Period. Mat. (4) 5, 80-85, 1925.

Cite this as:

Weisstein, Eric W. "Padoa's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PadoasInequality.html

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