Erdos–Mordell inequality
Encyclopedia
In Euclidean geometry
, the Erdős–Mordell inequality states that for any triangle ABC and point O inside ABC, the sum of the distances from O to the sides is less than or equal to half of the sum of the distances from O to the vertices. It is named after Paul Erdős
and Louis Mordell
. posed the problem of proving the identity; a proof was provided two years later by . This solution was however not very elementary. Subsequent simpler proofs were then found by , , and .
In absolute geometry
, the Erdős–Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is at most 2
.
Barrow's inequality
is a strengthened version of the Erdős–Mordell inequality in which the distances from O to the sides are replaced by the distances from O to the points where the angle bisectors cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, the Erdős–Mordell inequality states that for any triangle ABC and point O inside ABC, the sum of the distances from O to the sides is less than or equal to half of the sum of the distances from O to the vertices. It is named after Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and Louis Mordell
Louis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
. posed the problem of proving the identity; a proof was provided two years later by . This solution was however not very elementary. Subsequent simpler proofs were then found by , , and .
In absolute geometry
Absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...
, the Erdős–Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is at most 2
.Barrow's inequality
Barrow's inequality
In geometry, Barrow's inequality states the following: Let P be a point inside the triangle ABC; U, V, and W be the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively...
is a strengthened version of the Erdős–Mordell inequality in which the distances from O to the sides are replaced by the distances from O to the points where the angle bisectors cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.
External links
- Alexander Bogomolny, "Erdös-Mordell Inequality", from Cut-the-KnotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
.