Weitzenböck's inequality

For a triangle with side lengths a, b, and c and area K, it states that a² + b² + c² ≥ 4√3K, with equality exactly for an equilateral triangle.

Heron's formula: A formula expressing a triangle's area from its three side lengths. Substituting its area expression turns the inequality into an algebraic condition on the sides.
Triangle: A polygon with three sides and three interior angles. The inequality is a universal constraint on the measurements of this basic plane figure.
Hadwiger–Finsler inequality: A triangle inequality relating its area, inradius, circumradius, and side lengths. It combines area and side data in a different bound with a different geometric interpretation.
Schur's inequality: A symmetric polynomial inequality relating sums of powers of nonnegative variables. A Schur inequality can establish the side-length algebra behind a proof.
Triangle inequality: The condition that each side of a triangle is shorter than the sum of the other two. The inequality becomes especially transparent after expressing sides through positive semiperimeter differences.
Triangle area: The amount of plane region enclosed by a triangle, commonly denoted K. The area is the geometric quantity bounded by the sum of squared sides.
Euler's inequality: The inequality R ≥ 2r for a triangle's circumradius R and inradius r. Unlike the squared-side area bound, it compares two circle radii and shares equality at equilateral triangles.
Olympiad geometry: A style of mathematical problem solving focused on challenging geometric problems and proofs. The inequality supplies a concise tool in triangle area and side-length problems.
Semiperimeter: Half the sum of a polygon's side lengths; for a triangle, it is denoted s. The factors s−a, s−b, and s−c in Heron's formula help prove the bound.
Triangle perimeter: The sum of the lengths of a triangle's three sides. Perimeter-based inequalities provide another way to compare the same side data with area.

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Erdős–Mordell inequalityCompared with: It bounds side lengths and area, rather than distances from an interior point.