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.
From the cosines law we have: α being the angle between b and c. This can be transformed into: Since A=1/2bcsinα we have: Now remember that and Using this we get: Doing this for all sides of the triangle and adding up we get: β and γ being the other angles of the triangle.
Now since the halves of the triangle’s angles are less than π/2 the function tan is convex we have: Using this we get: This is the Hadwiger-Finsler inequality.
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.