View source for Hadwiger–Finsler inequality

In [[mathematics]], the '''Hadwiger–Finsler inequality''' is a result on the [[geometry]] of [[triangle]]s in the [[Euclidean plane]]. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then

:<math>a^{2} + b^{2} + c^{2} \geq (a - b)^{2} + (b - c)^{2} + (c - a)^{2} + 4 \sqrt{3} T \quad \mbox{(HF)}.</math>

==Related inequalities==
* [[Weitzenböck's inequality]] is a straightforward [[corollary]] of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then

:<math>a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3} T \quad \mbox{(W)}.</math>

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]] the triangle is an [[equilateral triangle]], i.e. ''a''&nbsp;=&nbsp;''b''&nbsp;=&nbsp;''c''.

* A version for [[quadrilateral]]: Let ''ABCD'' be a convex quadrilateral with the lengths ''a'', ''b'', ''c'', ''d'' and the area ''T'' then:<ref>Leonard Mihai Giugiuc, Dao Thanh Oai and Kadir Altintas, ''An inequality related to the lengths and area of a convex quadrilateral'', International Journal of Geometry, Vol. 7 (2018), No. 1, pp. 81 - 86, [https://ijgeometry.com/wp-content/uploads/2018/04/81-86.pdf]</ref>

:<math> a^2+b^2+c^2+d^2 \ge 4T + \frac{\sqrt{3}-1}{\sqrt{3}}\sum{(a-b)^2} </math> with equality only for a [[square]]. 
Where <math> \sum{(a-b)^2}=(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2 </math>

==History==
The Hadwiger–Finsler inequality is named after {{harvs|first1=Paul|last1=Finsler|author1-link=Paul Finsler|first2=Hugo|last2=Hadwiger|author2-link=Hugo Hadwiger|year=1937|txt}}, who also published in the same paper the [[Finsler–Hadwiger theorem]] on a square derived from two other squares that share a vertex.

==See also==
*[[List of triangle inequalities]]
*[[Isoperimetric inequality]]

==References==
{{Reflist}}
*{{cite journal|last1 = Finsler|first1 = Paul | author1-link = Paul Finsler | last2=Hadwiger | first2 = Hugo | author2-link = Hugo Hadwiger | title = Einige Relationen im Dreieck|journal = [[Commentarii Mathematici Helvetici]]|volume = 10|issue = 1|year = 1937|pages = 316–326|doi = 10.1007/BF01214300 | ref=harv}}
*Claudi Alsina, Roger B. Nelsen: ''When Less is More: Visualizing Basic Inequalities''. MAA, 2009, {{ISBN|9780883853429}}, pp. [https://books.google.de/books?id=U1ovBsSRNscC&pg=PA84 84-86]

==External links==
* {{PlanetMath|urlname=proofofhadwigerfinslerinequality|title=Proof of Hadwiger-Finsler inequality}}
* {{PlanetMath|urlname=WeizenbocksInequality|title=Weizenbock's inequality}}

{{DEFAULTSORT:Hadwiger-Finsler inequality}}
[[Category:Euclidean geometry]]
[[Category:Geometric inequalities]]
[[Category:Triangle geometry]]