Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Geometry

Thales’ Theorem

Fix a sphere and two points already known to be endpoints of one of its diameters. For any point p₂, the angle from the first endpoint through p₂ to the other endpoint is a right angle if and only if p₂ lies on the sphere.

Exact theorem

Exact Mathlib statement

theorem EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} {s : EuclideanGeometry.Sphere P} (hd : s.IsDiameter p₁ p₃) : ∠ p₁ p₂ p₃ = π / 2 ↔ p₂ ∈ s

The theorem at a glance

Thales’ Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

For fixed diameter endpoints p₁ and p₃, a point p₂ lies on the sphere if and only if the angle at p₂ is π/2. Explanatory diagram.
Detailed visual description

The poster pairs the exact equivalence with a single spherical boundary: p₁ and p₃ are antipodal endpoints of a centered diameter, p₂ lies on the same boundary, and its two chords meet at a right-angle marker. The footer preserves the arbitrary real inner-product affine-space scope and the absence of distinctness or positive-radius hypotheses.

Statement structure

Statement and scope

Statement map for Thales’ TheoremGiven two endpoints of a diameter of a sphere in a real inner-product affine space, the angle they subtend at a point is right exactly when that point lies on the sphere. Claim boundary: This page indexes Mathlib’s theorem for a metric affine space P modeled on a real inner-product space V. Once s.IsDiameter p₁ p₃ is given, the declaration proves ∠p₁p₂p₃ = π/2 if and only if p₂ ∈ s. It has no dimension-two, pairwise-distinctness, or positive-radius hypothesis. A planar great-circle picture is an illustrative cross-section, not the theorem’s full scope. The selected declaration does not prove the nearby two-dimensional result that recovers a diameter from three points already lying on an arbitrary sphere. The pinned upstream declaration is EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingGiven two endpoints of adiameter of a sphere ina real inner-productaffine space, the anglethey subtend at a pointis right exactly whenthat point lies on thesphere.Claim boundaryThis page indexesMathlib’s theorem for ametric affine space Pmodeled on a realinner-product space V.Once s.IsDiameter p₁ p₃is given, thedeclaration proves∠p₁p₂p₃ = π/2 if andonly if p₂ ∈ s. It hasno dimension-two,pairwise-distinctness,or positive-radiushypothesis. A planargreat-circle picture isan illustrativecross-section, not thetheorem’s full scope.The selected declarationdoes not prove thenearby two-dimensionalresult that recovers adiameter from threepoints already lying onan arbitrary sphere.Pinned declarationmathlib ·EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameterStatement map for Thales’ TheoremGiven two endpoints of a diameter of a sphere in a real inner-product affine space, the angle they subtend at a point is right exactly when that point lies on the sphere. Claim boundary: This page indexes Mathlib’s theorem for a metric affine space P modeled on a real inner-product space V. Once s.IsDiameter p₁ p₃ is given, the declaration proves ∠p₁p₂p₃ = π/2 if and only if p₂ ∈ s. It has no dimension-two, pairwise-distinctness, or positive-radius hypothesis. A planar great-circle picture is an illustrative cross-section, not the theorem’s full scope. The selected declaration does not prove the nearby two-dimensional result that recovers a diameter from three points already lying on an arbitrary sphere. The pinned upstream declaration is EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingGiven two endpoints of adiameter of a sphere ina real inner-productaffine space, the anglethey subtend at a pointis right exactly whenthat point lies on thesphere.Claim boundaryThis page indexesMathlib’s theorem for ametric affine space Pmodeled on a realinner-product space V.Once s.IsDiameter p₁ p₃is given, thedeclaration proves∠p₁p₂p₃ = π/2 if andonly if p₂ ∈ s. It hasno dimension-two,pairwise-distinctness,or positive-radiushypothesis. A planargreat-circle picture isan illustrativecross-section, not thetheorem’s full scope.The selected declarationdoes not prove thenearby two-dimensionalresult that recovers adiameter from threepoints already lying onan arbitrary sphere.Pinned declarationmathlib ·EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

With p₁ and p₃ fixed as diameter endpoints, the angle at p₂ is right exactly when p₂ lies on the sphere. Explanatory scientific diagram.
Detailed visual description

A flat emerald circle on warm technical paper provides a planar great-circle cross-section. The navy diameter joins p₁ at left to p₃ at right and passes through the antique-gold center c. The upper boundary point p₂ is connected to both endpoints, and a gold square marks the right angle at p₂. The picture is an illustrative cross-section of the arbitrary real inner-product affine-space theorem, not a dimension-two restriction.

Why it matters

A mathematical landmark

Thales’ theorem is a foundational bridge between incidence and orthogonality. Mathlib states the result as an exact two-way sphere-membership equivalence in an arbitrary real inner-product affine space, and its proof exposes the inner-product identity behind the familiar great-circle picture.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib’s theorem for a metric affine space P modeled on a real inner-product space V. Once s.IsDiameter p₁ p₃ is given, the declaration proves ∠p₁p₂p₃ = π/2 if and only if p₂ ∈ s. It has no dimension-two, pairwise-distinctness, or positive-radius hypothesis. A planar great-circle picture is an illustrative cross-section, not the theorem’s full scope. The selected declaration does not prove the nearby two-dimensional result that recovers a diameter from three points already lying on an arbitrary sphere.

Source and local evidence

Where the theorem comes from

Existing declaration
EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.thales-theorem.v001
Source
Open the pinned upstream reference