Pedoe's inequality
Encyclopedia
In geometry
, Pedoe's inequality, named after Daniel Pedoe
, states that if a, b, and c are the lengths of the sides of a triangle
with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then
with equality if and only if
the two triangles are similar
.
The expression on the left is not only symmetric under any of the six permutations of the set { (A, a), (B, b), (C, c) } 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
and of the Hadwiger–Finsler inequality
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, Pedoe's inequality, named after Daniel Pedoe
Daniel Pedoe
Dan Pedoe 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 expository papers in geometry. He is also the author of various core books on mathematics and geometry some of which have...
, states that if a, b, and c are the lengths of the sides of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then
with equality if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the two triangles are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
.
The expression on the left is not only symmetric under any of the six permutations of the set { (A, a), (B, b), (C, c) } 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
Weitzenböck's inequality
In mathematics, Weitzenböck's inequality states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds:Equality occurs if and only if the triangle is equilateral...
and of the Hadwiger–Finsler inequality
Hadwiger–Finsler inequality
In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler...
.