Weitzenböck inequality

{\ displaystyle {\ begin {aligned} & {\ text {all interior angles}} <120 ^ {\ circ}: \\ & {\ text {gray area}} = 3F \ leq F_ {a} + F_ {b} + F_ {c} \ end {aligned}}}

{\ displaystyle {\ begin {aligned} & {\ text {an interior angle}} \ geq 120 ^ {\ circ}: \\ & {\ text {gray area}} = 3F \ leq F_ {c} <F_ {a } + F_ {b} + F_ {c} \ end {aligned}}}

In the mathematics that states inequality Weitzenböck , named after Roland Weitzenböck that for a triangle ABC with sides , , and its area is considered the following statement: ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle F}$

{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} \ geq 4 {\ sqrt {3}} \, F}

.

Equality applies if and only if the triangle is equilateral. The inequality of Pedoe is a generalization of the inequality of Weitzenböck two triangles. A tightening of Weitzenböck's inequality is provided by Hadwiger-Finsler's inequality .

Geometric interpretation and evidence

If you rearrange the equation a bit, you get a representation whose geometrical interpretation results in a simple geometrical proof.

{\ displaystyle {\ frac {\ sqrt {3}} {4}} a ^ {2} + {\ frac {\ sqrt {3}} {4}} b ^ {2} + {\ frac {\ sqrt { 3}} {4}} c ^ {2} \ geq 3F}

.

Here on the left side of the inequality is the sum of the areas of the equilateral triangles that have been built over the sides of triangle ABC, so this is always greater than or equal to three times the area of triangle ABC.

{\ displaystyle F_ {a} + F_ {b} + F_ {c} \ geq 3F}

.

If the triangle ABC is now broken down at its Fermat point P into three partial triangles, each of these partial triangles has an -angle in P and fits three times into the adjacent equilateral triangle. In this way, three times the area of triangle ABC has been generated within the equilateral triangles, the sum of which must therefore be greater or equal. This decomposition of the triangle ABC is only possible if all of its interior angles are smaller than . If this is not the case, the entire triangle can be placed three times within the equilateral triangle over the longest side, which means that the total area of all equilateral triangles is even greater. ${\ displaystyle 120 ^ {\ circ}}$ ${\ displaystyle 120 ^ {\ circ}}$

Proof with Heron's formula

With the help of the Heronian formula for the triangular surfaces one obtains an algebraic proof:

{\ displaystyle {\ begin {aligned} F & {} = {\ frac {\ sqrt {(a + b + c) (a + bc) (b + ca) (c + ab)}} {4}} \\ & {} = {\ frac {1} {4}} {\ sqrt {2 (a ^ {2} b ^ {2} + a ^ {2} c ^ {2} + b ^ {2} c ^ { 2}) - (a ^ {4} + b ^ {4} + c ^ {4})}}. \ End {aligned}}}

From the nonnegativity of squares it follows:

{\ displaystyle {\ begin {aligned} {} & (a ^ {2} -b ^ {2}) ^ {2} + (b ^ {2} -c ^ {2}) ^ {2} + (c ^ {2} -a ^ {2}) ^ {2} \ geq 0 \\ {} \ iff & 2 (a ^ {4} + b ^ {4} + c ^ {4}) - 2 (a ^ { 2} b ^ {2} + a ^ {2} c ^ {2} + b ^ {2} c ^ {2}) \ geq 0 \\ {} \ iff & {\ frac {4 (a ^ {4 } + b ^ {4} + c ^ {4})} {3}} \ geq {\ frac {4 (a ^ {2} b ^ {2} + a ^ {2} c ^ {2} + b ^ {2} c ^ {2})} {3}} \\ {} \ iff & {\ frac {(a ^ {4} + b ^ {4} + c ^ {4}) + 2 (a ^ {2} b ^ {2} + a ^ {2} c ^ {2} + b ^ {2} c ^ {2})} {3}} \ geq 2 (a ^ {2} b ^ {2} + a ^ {2} c ^ {2} + b ^ {2} c ^ {2}) - (a ^ {4} + b ^ {4} + c ^ {4}) \\ {} \ iff & {\ frac {(a ^ {2} + b ^ {2} + c ^ {2}) ^ {2}} {3}} \ geq (4F) ^ {2} \ end {aligned}}}

The assertion follows immediately from this if one takes the square root on both sides . The first line also shows that equality occurs exactly for , i.e. for equilateral triangles. ${\ displaystyle a = b = c}$

history

The inequality was first published in the Romanian mathematics journal Gazetei Matematice on 1897 . There, Ion Ionescu, the editor of the magazine, set the following task (problem 273):

"Show that there is no triangle for which applies" ${\ displaystyle 4F {\ sqrt {3}}> a ^ {2} + b ^ {2} + c ^ {2}}$

- Ion Ionescu

A solution to the problem was published by the magazine the following year. Independently of this, Roland Weitzenböck published an article in the mathematical journal in 1919 with the title About an inequality in triangular geometry . There he presented the following inequality along with several proofs and extensions:

"If F is the area of a flat triangle with sides a, b and c, then is always ." ${\ displaystyle {\ tfrac {F} {a ^ {2} + b ^ {2} + c ^ {2}}} \ leq {\ tfrac {1} {12}} {\ sqrt {3}}}$

- Roland Weitzenböck

In 1961 the inequality was used for the second task in the third International Mathematical Olympiad in Veszprém ( Hungary ):

“Let a, b, c be the sides of a triangle and K its surface. Evidence . In which case does the equality apply? " ${\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} \ geq 4 {\ sqrt {3}} K}$

- Task 2 in the 3rd International Mathematical Olympiad

literature

Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities . MAA, 2009, ISBN 978-0-88385-342-9 , pp. 84-86
Claudi Alsina, Roger B. Nelsen: Geometric Proofs of the Weitzenböck and Hadwiger-Finsler Inequalities . In: Mathematics Magazine , Vol. 81, No. 3 (Jun., 2008), pp. 216-219 ( JSTOR 27643111 )
DM Batinetu-Giurgiu, Nicusor Minculete, Nevulai Stanciu: Some geometric inequalities of Ionescu-Weitzebböck type . In: International Journal of Geometry , Vol. 2 (2013), No. April 1
DM Batinetu-Giurgiu, Nevulai Stanciu: The inequality Ionescu - Weitzenböck . MateInfo.ro, April 2013, ( mateinfo.ro )
Daniel Pedoe : On Some Geometrical Inequalities . In: The Mathematical Gazette , Vol. 26, No. 272, December 1942, pp. 202-208 ( JSTOR 3607041 )
Roland Weitzenböck : About an inequality in triangular geometry . In: Mathematische Zeitschrift , Volume 5, 1919, pp. 137–146 ( digitized at the Göttingen Digitization Center )
Dragutin Svrtan, Darko Veljan: Non-Euclidean Versions of Some Classical Triangle Inequalities . In: Forum Geometricorum , Volume 12, 2012, pp. 197-209; forumgeom.fau.edu (PDF)
Mihaly Bencze, Nicusor Minculete, Ovidiu T. Pop: New inequalities for the triangle . In: Octogon Mathematical Magazine , Vol. 17, No. 1, April 2009, pp. 70-89; uni-miskolc.hu (PDF)

Web links

Weitzenböck's Inequality on cut-the-knot.org with animated representation
Eric W. Weisstein : Weitzenböck's Inequality . In: MathWorld (English).

Individual evidence

^ Claudi Alsina, Roger B. Nelsen: Geometric Proofs of the Weitzenböck and Hadwiger-Finsler Inequalities . In: Mathematics Magazine , Vol. 81, No. 3, June 2008, pp. 216-219 ( JSTOR 27643111 )
↑ Original publication : Gazeta Matematică, Vol. III (September 15, 1897 - August 15, 1898), No. 2, Octombrie 1897, la pagina 52. See also: DM Batinetu-Giurgiu, Nevulai Stanciu: Inegalitatea Ionescu - Weitzenböck . In: Gazeta Matematica-Seria B , ° 118 (1), 2013, pp. 1-10; ssmr.ro (PDF)
↑ Roland Weitzenböck : About an inequality in the triangular geometry . In: Mathematische Zeitschrift , Volume 5, 1919, pp. 137–146 ( digitized at the Göttingen Digitization Center )
^ Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . MAA 2010, ISBN 978-0-88385-348-1 , pp. 96-98 ( excerpt (Google) )

[1] Claudi Alsina, Roger B. Nelsen: Geometric Proofs of the Weitzenböck and Hadwiger-Finsler Inequalities . In: Mathematics Magazine , Vol. 81, No. 3, June 2008, pp. 216-219 ( JSTOR 27643111 )

[2] Original publication : Gazeta Matematică, Vol. III (September 15, 1897 - August 15, 1898), No. 2, Octombrie 1897, la pagina 52. See also: DM Batinetu-Giurgiu, Nevulai Stanciu: Inegalitatea Ionescu - Weitzenböck . In: Gazeta Matematica-Seria B , ° 118 (1), 2013, pp. 1-10; ssmr.ro (PDF)

[3] Roland Weitzenböck : About an inequality in the triangular geometry . In: Mathematische Zeitschrift , Volume 5, 1919, pp. 137–146 ( digitized at the Göttingen Digitization Center )

[4] Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . MAA 2010, ISBN 978-0-88385-348-1 , pp. 96-98 ( excerpt (Google) )