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 , and and area , then
Related inequalities Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lendths , and and area , then
Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF)Martin Lukarevski, The circummidarc triangle and the Finsler-Hadwiger inequality, Math. Gaz. 104 (July 2020) pp. 335-338. doi:10.1017/mag.2020.63
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 version for quadrilateral: Let be a convex quadrilateral with the lengths and the area then: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,
with equality only for a square. Where
Proof From the cosines law we have:
being the angle between and . This can be transformed into:
Since we have:
Now remember that
and
Using this we get:
Doing this for all sides of the triangle and adding up we get:
and being the other angles of the triangle. Now since the halves of the triangle’s angles are less than the function is convex we have:
Using this we get:
This is the Hadwiger-Finsler inequality. History The Hadwiger–Finsler inequality is named after , 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
Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, , pp. 84-86
External links