Knowra Erdős–Mordell inequality Erdős–Mordell inequality For any point inside a triangle, the sum of its distances to the three vertices is at least twice the sum of its perpendicular distances to the three sides.
Triangle : A polygon with three sides, three vertices, and three interior angles. The inequality compares distances within this basic geometric figure.
Paul Erdős : A Hungarian mathematician known for prolific collaboration and contributions to number theory and combinatorics. He posed the inequality that bears his name.
Distance geometry : The study of geometric structure through distances between points. The inequality is a concrete constraint on distances in a planar configuration.
Weitzenböck's inequality : For a triangle of area K and side lengths a, b, and c, a² + b² + c² ≥ 4√3K. It bounds side lengths and area, rather than distances from an interior point.
Point-to-line distance : The shortest distance from a point to a line, measured along a perpendicular. Each distance to a side in the inequality is a point-to-line distance.
Louis Mordell : A British mathematician known for work in number theory, including the Mordell theorem. He independently posed the same triangle inequality with Erdős.
Olympiad geometry : Problem-solving geometry used in mathematical competitions, emphasizing synthetic arguments and clever constructions. The result and its variants appear in competition-style inequality problems.
Euler's triangle inequality : For a triangle's circumradius R and inradius r, R ≥ 2r. It compares two global triangle radii, not point-to-vertex and point-to-side distances.
Triangle inequality : A geometric principle stating that one side of a triangle is shorter than the sum of the other two. Proofs use triangle inequalities to relate vertex distances to side distances.
Geometric inequality : An inequality relating numerical quantities defined by geometric objects. The result belongs to a broad tradition of bounding geometric measurements.
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