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

Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF)[1]

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 = b = c.

  • A version for quadrilateral: Let ABCD be a convex quadrilateral with the lengths a, b, c, d and the area T then:[2]
with equality only for a square.

Where

History

The Hadwiger–Finsler inequality is named after Paul Finsler and Hugo Hadwiger (1937), 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

References

  1. Martin Lukarevski, The circummidarc triangle and the Finsler-Hadwiger inequality, Math. Gaz. 104 (July 2020) pp. 335-338. doi:10.1017/mag.2020.63
  2. 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,
  • Finsler, Paul; Hadwiger, Hugo (1937). "Einige Relationen im Dreieck". Commentarii Mathematici Helvetici. 10 (1): 316–326. doi:10.1007/BF01214300.CS1 maint: ref=harv (link)
  • Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, ISBN 9780883853429, pp. 84-86
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.