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The 51 pages that link to Triangle inequality, each with the reason it gives.

• TriangleRelated: It identifies which three proposed lengths can form a triangle.

• Metric spaceRelated: It constrains distances so detours cannot be shorter than direct separation.

• Absolute valueRelated: For real numbers, it gives the bound |x+y| ≤ |x|+|y|.

• Cauchy–Schwarz inequalityRelated: The inner-product bound is a standard step in proving the norm triangle inequality.

• Equilateral triangleRelated: Equal positive side lengths satisfy this condition and form a nondegenerate triangle.

• Heron's formulaRelated: It ensures the side lengths describe a nondegenerate triangle with positive area.

• Law of cosinesRelated: The law’s side lengths must still satisfy these bounds to form a triangle.

• DistanceRelated: This constraint is central to the mathematical definition of distance.

• Cauchy criterionRelated: It connects pairwise closeness to closeness around a proposed limit.

• Normed vector spaceRelated: For a norm, it bounds the size of a sum by the sizes of its summands.

• IncenterRelated: The inradius formula uses the triangle’s side lengths and area, which must satisfy this condition.

• Euclidean normRelated: The Euclidean norm satisfies this defining length bound for vector addition.

• Isosceles triangleRelated: It limits which side lengths can form an isosceles triangle.

• SemiperimeterRelated: It ensures each difference s−a, s−b, and s−c is positive for a nondegenerate triangle.

• Clebsch–Gordan coefficientsRelated: The allowed total angular momentum must satisfy the corresponding triangle rule.

• Brahmagupta's formulaRelated: Side lengths must also satisfy polygon feasibility constraints before the formula describes a real figure.

• Complex modulusRelated: Modulus obeys this inequality for sums of complex numbers.

• Euler's inequalityRelated: Its side-length constraints help establish the algebraic inequality behind Euler's result.

• Parallelogram lawRelated: It is another core norm property, though it does not imply the parallelogram identity.

• Schur's inequalityRelated: Substituting triangle side lengths into Schur-type bounds can yield useful relations among their sums and products.

• Ptolemy's inequalityRelated: A standard proof reduces the quadrilateral bound to triangle inequalities after a suitable construction.

• Scalene triangleRelated: It limits which three different lengths can form a scalene triangle.

• Magnitude (mathematics)Related: Norm-based magnitudes obey this bound, which constrains sums and distances.

• Stewart's theoremRelated: The lengths in Stewart’s theorem must also satisfy the triangle’s basic feasibility constraints.

• Weitzenböck's inequalityRelated: The inequality becomes especially transparent after expressing sides through positive semiperimeter differences.

• Hadwiger–Finsler inequalityRelated: Its constraints ensure the side lengths used in the inequality form a genuine triangle.

• Travelling salesman problemRelated: Metric tours satisfy this condition, enabling stronger approximation guarantees.

• Estimation lemmaRelated: Its integral form bounds the modulus of an integral by the integral of the modulus.

• Fagnano's problemRelated: It certifies that the straightened reflected path gives a perimeter lower bound.

• Gershgorin circle theoremRelated: Applying it to an eigenvector equation produces the row-sum radius.

• Midpoint theoremRelated: The half-length conclusion is consistent with the bounds imposed on every triangle’s sides.

• Abel's inequalityRelated: After the transform, it bounds the remaining terms by absolute values.

• Barrow's inequalityRelated: Proofs use it to bound vertex distances through points on the sides.

• Equal Incircles TheoremRelated: It restricts which side-length triples can satisfy the theorem’s conditions.

• Erdős–Mordell inequalityRelated: Proofs use triangle inequalities to relate vertex distances to side distances.

• Hinge theoremRelated: Its bounds help explain why changing the included angle changes the third side.

• Incenter–excenter lemmaRelated: It constrains the side-length expressions used in center-distance results.

• Inverse Pythagorean theoremRelated: Side lengths must form a triangle before the converse can classify its angle.

• Mollweide's formulaRelated: The side-sum and side-difference ratios are consistent with these basic bounds.

• Peetre's inequalityRelated: The Euclidean triangle inequality supplies the basic comparison behind the polynomial weight estimate.

• Pompeiu's theoremRelated: After rotating one vertex-to-point segment by 60°, this inequality bounds the third distance by the other two.

• Riesz's lemmaRelated: It bounds the chosen vector's norm using its distance from the subspace.

• Steiner–Lehmus theoremRelated: Proofs use valid triangle side constraints to rule out extraneous algebraic possibilities.