Linked from

The 63 pages that link to Pythagorean theorem, each with the reason it gives.

• Euclidean geometryBroader topic: It converts perpendicular side lengths into straight-line distances.

• TriangleBroader topic: This special side-length relation applies when one of the triangle’s angles is a right angle.

• Euclidean spaceRelated: Orthogonal displacement components combine by this rule to give squared distance.

• Euclid's ElementsBroader topic: Euclid gives a deductive proof of this result in Book I.

• Euclidean distanceRelated: Repeated applications of this theorem yield the coordinate formula for distance.

• Equilateral triangleRelated: Splitting the triangle by an altitude gives a right triangle for deriving its height.

• Inner productRelated: Orthogonal vectors satisfy this length-addition law under an inner-product norm.

• Heron's formulaRelated: A perpendicular height and the triangle's sides form right triangles in a standard derivation.

• Right triangleRelated: It gives the defining length relationship among the triangle’s three sides.

• Law of cosinesCompared with: When the included angle is 90°, the law reduces exactly to this theorem.

• PythagorasBroader topic: Later tradition attached this theorem to Pythagoras, though its earlier history is uncertain.

• Square rootRelated: Finding a side length from the other two requires taking a square root.

• DistanceRelated: It yields the familiar distance formula by combining perpendicular coordinate differences.

• OrthogonalityRelated: Orthogonal vectors satisfy the same sum-of-squares relation for their lengths.

• TrigonometryRelated: It links the sine and cosine of an angle through their squared values.

• Unit circleRelated: For unit-circle coordinates, it yields cos² θ + sin² θ = 1.

• PythagoreanismBroader topic: Its traditional attribution illustrates the community’s enduring association with mathematics.

• Orthogonal projectionRelated: Perpendicularity makes projection error split into a right-triangle distance identity.

• DisplacementRelated: For perpendicular components, it gives displacement magnitude.

• Euclidean normRelated: The norm formula extends this theorem from two perpendicular components to any number of coordinates.

• Isosceles triangleRelated: The altitude splits the triangle into right triangles whose side lengths can be calculated with this theorem.

• Power of a pointRelated: A perpendicular from the center to a secant derives the center-distance formula.

• Babylonian mathematicsRelated: Babylonian numerical problems include right-triangle relations centuries before Pythagoras.

• Greek mathematicsBroader topic: Its proof and geometric consequences became central to Greek number and shape studies.

• Perpendicular linesRelated: Perpendicular sides form the legs in the theorem's right-triangle applications.

• RectangleRelated: It gives a rectangle’s diagonal length from its side lengths.

• TheoremBroader topic: Its many proofs make a precise geometric relation a classic example of a theorem.

• Inner product spaceRelated: Orthogonal vectors satisfy this identity through the inner-product norm.

• SemiperimeterCompared with: Unlike the semiperimeter, it relates side lengths through squares rather than their sum.

• SquareRelated: Applied to a square’s diagonal, it gives the diagonal-to-side relation.

• Chinese mathematicsRelated: Chinese texts applied the right-triangle relation to surveying and geometric problems.

• Thales' theoremRelated: Thales' theorem identifies right triangles to which this relation applies.

• Euclidean planeRelated: It yields the distance formula for points in perpendicular coordinates.

• Pythagorean tripleNarrower topic: The triple equation is the integer-side specialization of this geometric theorem.

• Right angleRelated: Its converse can establish that a triangle contains a right angle.

• Algebraic identityRelated: Its equation holds for every right triangle, linking geometric relations to algebraic equality.

• Complex modulusRelated: Applied to the real and imaginary coordinates, it gives the modulus formula.

• Distance formulaRelated: The distance formula applies this theorem to horizontal and vertical coordinate differences.

• Law of sinesCompared with: It solves right triangles through side lengths, while the law of sines applies to triangles of any shape.

• Geometric proofBroader topic: Its many proofs illustrate distinct ways to establish one geometric claim.

• Parallelogram lawCompared with: It is the right-angle special case behind the law’s diagonal-length relation.

• Perfect squareRelated: Its integer solutions connect perfect squares through sums of two squares.

• The Nine Chapters on the Mathematical ArtRelated: The gougu chapter applies this relation to inaccessible lengths and right-triangle problems.

• Unit sphereRelated: Applied along three perpendicular axes, it gives the sphere's equation.

• Bessel's inequalityRelated: The inequality follows by applying this relation to a vector and its orthogonal projection.

• Fermat's theorem on sums of two squaresRelated: The same sum-of-squares expression appears, though the theorem concerns prime hypotenuses.

• Hippocrates of ChiosRelated: Right triangles provide the side-length relationships used in classic lune constructions.

• Sum of squaresRelated: Orthogonal fitted and residual components yield an analogous sum-of-squares decomposition.

• Van Aubel's theoremRelated: Length comparisons in geometric proofs often reduce to this relation.

• Nth rootRelated: Recovering a side length from the other two requires taking a square root.