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 ...

, Pedoe's inequality (also Neuberg–Pedoe inequality), named after

Daniel Pedoe Dan Pedoe (29 October 1910, London – 27 October 1998, St Paul, Minnesota, USA) was an English-born mathematician and geometer with a career spanning more than sixty years. In the course of his life he wrote approximately fifty research and e ...

(1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a

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 ...

with area ''ƒ'', and ''A'', ''B'', and ''C'' are the lengths of the sides of another triangle with area ''F'', then :

A^2(b^2+c^2-a^2)+B^2(a^2+c^2-b^2)+C^2(a^2+b^2-c^2)\geq 16Ff,\,

with equality

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 two triangles are similar with pairs of corresponding sides (''A, a''), (''B, b''), and (''C, c''). The expression on the left is not only symmetric under any of the six

permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...

of the set of pairs, but also—perhaps not so obviously—remains the same if ''a'' is interchanged with ''A'' and ''b'' with ''B'' and ''c'' with ''C''. In other words, it is a symmetric function of the pair of triangles. Pedoe's inequality is a generalization of Weitzenböck's inequality, which is the case in which one of the triangles is

equilateral 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 ...

. Pedoe discovered the inequality in 1941 and published it subsequently in several articles. Later he learned that the inequality was already known in the 19th century to Neuberg, who however did not prove that the equality implies the similarity of the two triangles.

Proof

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 ...

, the area of the two triangles can be expressed as: :

16f^2=(a+b+c)(a+b-c)(a-b+c)(b+c-a)=(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)

16F^2=(A+B+C)(A+B-C)(A-B+C)(B+C-A)=(A^2+B^2+C^2)^2-2(A^4+B^4+C^4)

and then, using Cauchy-Schwarz inequality we have, :

16Ff+2a^2A^2+2b^2B^2+2c^2C^2

\leq \sqrt\sqrt

= (a^2+b^2+c^2)(A^2+B^2+C^2)

So, :

16Ff\leq  A^2(a^2+b^2+c^2)-2a^2A^2+B^2(a^2+b^2+c^2)-2b^2B^2+C^2(a^2+b^2+c^2)-2c^2C^2

=A^2(b^2+c^2-a^2)+B^2(a^2+c^2-b^2)+C^2(a^2+b^2-c^2)

and the proposition is proven. Equality holds if and only if

\tfrac=\tfrac=\tfrac=\sqrt

, that is, the two triangles are similar.

References

* * * * * * {{cite journal , last1=Poh , first1=K. S. , title=A short note on a Pedoe's theorem about two triangles , journal=Singapore Mathematical Society Mathematical Medley , volume=11 , issue=2 , url=https://sms.math.nus.edu.sg/smsmedley/Vol-11-2/A%20short%20note%20on%20a%20Pedoe%27s%20theorem%20about%20two%20triangles(KS%20Poh).pdf Triangle inequalities

Proof

See also

References