Weitzenböck's inequality
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Weitzenböck's inequality, named after
Roland Weitzenböck Roland Weitzenböck (26 May 1885 – 24 July 1955) was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection. He was appointed professor of mathematics at the University of Amsterdam in 1923 at the ...
, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt\, \Delta. Equality occurs if and only if the triangle is equilateral.
Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area ''& ...
is a generalization of Weitzenböck's inequality. The
Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ ...
is a strengthened version of Weitzenböck's inequality.


Geometric interpretation and proof

Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. : \fraca^2 + \fracb^2 + \fracc^2 \geq 3\, \Delta. Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of the original triangle. : \Delta_a + \Delta_b + \Delta_c \geq 3\, \Delta. This can now can be shown by replicating area of the triangle three times within the equilateral triangles. To achieve that the
Fermat point In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
is used to partition the triangle into three obtuse subtriangles with a 120^\circ angle and each of those subtriangles is replicated three times within the equilateral triangle next to it. This only works if every angle of the triangle is smaller than 120^\circ, since otherwise the Fermat point is not located in the interior of the triangle and becomes a vertex instead. However if one angle is greater or equal to 120^\circ it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow.


Further proofs

The proof of this inequality was set as a question in the
International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except i ...
of 1961. Even so, the result is not too difficult to derive using
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for the area of a triangle: : \begin \Delta & =\frac\sqrt \\ pt& =\frac\sqrt. \end


First method

It can be shown that the area of the inner
Napoleon's triangle In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral tr ...
, which must be nonnegative, isCoxeter, H.S.M., and Greitzer, Samuel L. ''Geometry Revisited'', page 64. :\frac(a^2+b^2+c^2-4\sqrt\Delta), so the expression in parentheses must be greater than or equal to 0.


Second method

This method assumes no knowledge of inequalities except that all squares are nonnegative. : \begin & (a^2 - b^2)^2 + (b^2 - c^2)^2 + (c^2 - a^2)^2 \geq 0 \\ pt \iff & 2(a^4+b^4+c^4) - 2(a^2 b^2+a^2c^2+b^2c^2) \geq 0 \\ pt \iff & \frac \geq \frac \\ pt \iff & \frac \geq 2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4) \\ pt \iff & \frac \geq (4\Delta)^2, \end 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.


Third method

This proof assumes knowledge of the
AM–GM inequality In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; a ...
. : \begin & & (a-b)^2+(b-c)^2+(c-a)^2 & \geq & & 0 \\ \Rightarrow & & 2a^2+2b^2+2c^2 & \geq & & 2ab+2bc+2ac \\ \iff & & 3(a^2+b^2+c^2) & \geq & & (a + b + c)^2 \\ \iff & & a^2+b^2+c^2 & \geq & & \sqrt \\ \Rightarrow & & a^2+b^2+c^2 & \geq & & \sqrt \\ \iff & & a^2+b^2+c^2 & \geq & & 4 \sqrt3 \Delta. \end As we have used the arithmetic-geometric mean inequality, equality only occurs when a = b = c and the triangle is equilateral.


Fourth method

Write x=\cot A, c=\cot A+\cot B>0 so the sum S=\cot A+\cot B+\cot C=c+\frac and cS=c^2-xc+x^2+1=\left(x-\frac\right)^2+\left(\frac-1\right)^2+c\sqrt\ge c\sqrt i.e. S\ge\sqrt. But \cot A=\frac, so S=\frac.


See also

*
List of triangle inequalities In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of th ...
*
Isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
*
Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ ...


Notes


References & further reading

*Claudi Alsina, Roger B. Nelsen: ''When Less is More: Visualizing Basic Inequalities''. MAA, 2009, , pp
84-86
*Claudi Alsina, Roger B. Nelsen: ''Geometric Proofs of the Weitzenböck and Hadwiger–Finsler Inequalities''. Mathematics Magazine, Vol. 81, No. 3 (Jun., 2008), pp. 216–219
JSTOR
*D. M. Batinetu-Giurgiu, Nicusor Minculete, Nevulai Stanciu
''Some geometric inequalities of Ionescu-Weitzebböck type''
International Journal of Geometry, Vol. 2 (2013), No. 1, April *D. M. Batinetu-Giurgiu, Nevulai Stanciu: ''The inequality Ionescu - Weitzenböck''. MateInfo.ro, April 2013,
online copy
*
Daniel Pedoe Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and expos ...
: ''On Some Geometrical Inequalities''. The Mathematical Gazette, Vol. 26, No. 272 (Dec., 1942), pp. 202-208
JSTOR
*
Roland Weitzenböck Roland Weitzenböck (26 May 1885 – 24 July 1955) was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection. He was appointed professor of mathematics at the University of Amsterdam in 1923 at the ...
: ''Über eine Ungleichung in der Dreiecksgeometrie''. Mathematische Zeitschrift, Volume 5, 1919, pp. 137-146
online copy
at
Göttinger Digitalisierungszentrum The Center for Retrospective Digitization in Göttingen (german: Göttinger DigitalisierungsZentrum, GDZ) is an online system for archiving academic journals maintained by the University of Göttingen The University of Göttingen, officially the ...
) *Dragutin Svrtan, Darko Veljan: ''Non-Euclidean Versions of Some Classical Triangle Inequalities''. Forum Geometricorum, Volume 12, 2012, pp. 197–209
online copy
*Mihaly Bencze, Nicusor Minculete, Ovidiu T. Pop: ''New inequalities for the triangle''. Octogon Mathematical Magazine, Vol. 17, No.1, April 2009, pp. 70-89
online copy


External links

*
Weitzenböck's Inequality
" an interactive demonstration by Jay Warendorff -
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