Pedoe's inequality

Pedoe's inequality states that for triangles with side lengths a, b, c and x, y, z and areas K and L, (a²+b²+c²)(x²+y²+z²) ≥ 48KL. Equality holds exactly when the triangles are similar.

Heron's formula: A formula giving a triangle's area from its three side lengths. It converts the areas in Pedoe's inequality into expressions involving the same side lengths.
Daniel Pedoe: A British mathematician known for work in geometry and the history of mathematics. The inequality bears Pedoe's name and is associated with his geometric work.
Isoperimetric inequality: A family of inequalities relating the boundary size of a shape to its enclosed area. Both results constrain area using measurements of a boundary, though Pedoe compares two triangles.
Hadwiger–Finsler inequality: A triangle inequality relating side lengths, area, and the circumradius. It constrains one triangle's measurements rather than pairing two triangles.
Triangle area: The measure of the planar region enclosed by a triangle. Pedoe's inequality compares each triangle's area with its squared side lengths.
Geometry of triangles: The study of properties and relations among triangles. Pedoe's inequality is one result in the broader tradition of triangle geometry.
Weitzenböck's inequality: For a triangle of area K and sides a, b, c, 4√3K ≤ a²+b²+c². It bounds one triangle's area, while Pedoe's inequality couples two triangles.
Euler's triangle inequality: The inequality R ≥ 2r for a triangle's circumradius R and inradius r. It concerns two radii of one triangle, unlike Pedoe's cross-triangle comparison.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. Similarity is precisely the equality case of Pedoe's inequality.
Geometric inequality: An inequality expressing a bound among quantities associated with geometric objects. Pedoe's result compares two triangles through areas and side lengths.

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