mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Hadwiger–Finsler inequality is a result on the

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s in the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :

a^ + b^ + c^ \geq (a - b)^ + (b - c)^ + (c - a)^ + 4 \sqrt T \quad \mbox.

Related inequalities

* Weitzenböck's inequality is a straightforward

corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...

of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :

a^ + b^ + c^ \geq 4 \sqrt T \quad \mbox.

Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF) Weitzenböck's inequality can also be proved using

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

, by which route it can be seen that equality holds in (W)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the triangle is an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

, i.e. ''a'' = ''b'' = ''c''. * A version for

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

: Let ''ABCD'' be a convex quadrilateral with the lengths ''a'', ''b'', ''c'', ''d'' and the area ''T'' 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

/ref> :

a^2+b^2+c^2+d^2 \ge 4T + \frac\sum

with equality only for a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

. Where

\sum=(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2

Proof

From the cosines law we have:

a^2=b^2+c^2-2bc\cos\alpha

α being the angle between b and c. This can be transformed into:

a^2=(b-c)^2+2bc(1-\cos\alpha)

Since A=1/2bcsinα we have:

a^2=(b-c)^2+4A\frac

Now remember that

1-\cos\alpha=2\sin^2\frac

and

\sin\alpha=2\sin\frac\cos\frac

Using this we get:

a^2=(b-c)^2+4A\tan\frac

Doing this for all sides of the triangle and adding up we get:

a^2+b^2+c^2=(a-b)^2+(b-c)^2+(c-a)^2+4A(\tan\frac+\tan\frac+\tan\frac)

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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

we have:

\tan\frac+\tan\frac+\tan\frac\geq3\tan\frac=3\tan\frac=\sqrt

Using this we get:

a^2 + b^2 + c^2 \geq (a-b)^2+(b-c)^2+(c-a)^2+ 4\sqrt\, A

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.

References

* *Claudi Alsina, Roger B. Nelsen: ''When Less is More: Visualizing Basic Inequalities''. MAA, 2009, , pp
84-86

External links

* * {{DEFAULTSORT:Hadwiger-Finsler inequality Triangle inequalities

Related inequalities

Proof

History

See also

References

External links