{ "artifactId": "artifact.library.mathlib.thales-theorem.v001", "candidateOnly": false, "category": "Geometry", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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", "family": "Euclidean sphere geometry", "id": "library.mathlib.thales-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/thales-theorem/", "entryData": "/data/library-theorems/thales-theorem.json", "evidence": "/proofs/artifact.library.mathlib.thales-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.thales-theorem.v001.evidence.json", "source": "/sources/upstream/thales-theorem/" }, "landmark": { "completeFormalType": "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", "concepts": [ "spheres", "diameters", "right angles", "affine inner-product spaces", "sphere membership" ], "plainLanguage": "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.", "poster": { "alt": "An ivory theorem poster states the fixed-diameter right-angle equivalence above a sphere cross-section whose three labeled boundary points form a right triangle.", "byteSize": 2720875, "caption": "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.", "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.", "derivatives": [ { "byteSize": 78312, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/thales-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:3f58e96ac4a65a637695925bcdb95771c5c53d8002c8691d032e487962348f72", "width": 640 }, { "byteSize": 201176, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/thales-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:7a6965d52fbed83cb9c8cf8ea60a69aff31c8c11e332493c1ea1ec1dec1b5184", "width": 1024 } ], "generatedAt": "2026-07-25T20:18:58.349Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/thales-theorem/theorem-poster-v1.png", "sha256": "sha256:8ce0b3a33bcacaa79b8b3d83420181843fe461bd43dade40143652824fc35d85", "title": "Thales’ Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "REAL INNER-PRODUCT AFFINE SPACES · SPHERES\nTHALES’ THEOREM\np₁ AND p₃ ARE DIAMETER ENDPOINTS OF s\n∠p₁p₂p₃ = π/2 ⇔ p₂ ∈ s\nRIGHT ANGLE ⇔ ON THE SPHERE\np₂\ns\np₁\np₃\nHOW TO READ IT\n1 · Fix the sphere s\n2 · Fix diameter endpoints p₁ and p₃\n3 · Choose any point p₂\n4 · Test the angle at p₂\nEXACT SCOPE\nArbitrary real inner-product affine space. No distinctness or positive-radius hypothesis." }, "visual": { "alt": "A spherical great-circle cross-section shows opposite diameter endpoints at left and right, a point on the top boundary, and two chords meeting there at a right-angle marker.", "byteSize": 2526509, "caption": "With p₁ and p₃ fixed as diameter endpoints, the angle at p₂ is right exactly when p₂ lies on the sphere.", "description": "A dark engraved sphere carries one centered horizontal diameter and a top boundary point joined to both endpoints. The three points form an exact right-triangle representative, while meridian arcs make clear that the planar cross-section illustrates the arbitrary real inner-product affine-space statement.", "derivatives": [ { "byteSize": 24228, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/thales-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:f50e3593f2a12a7cf8ab293e55a24f5a45369a82b40b27347da6072c17102260", "width": 640 }, { "byteSize": 81248, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/thales-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:a5e55b0a9a817cc7be3fff54cc1c6230dfb6cde4960480ad1983c464b19cd2f9", "width": 1200 } ], "generatedAt": "2026-07-25T20:18:58.349Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/thales-theorem/theorem-schematic-v1.png", "sha256": "sha256:fec8e808821ef0b8de120ea6d906f8d8ce06256e6c8988b24d09948e7f9ce0cd", "title": "A diameter, a sphere point, and a right angle", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "ProofAtlas did not originate Thales’ theorem or Mathlib’s declaration.", "The selected declaration is not restricted to planar circles or finite-dimensional spaces.", "The page does not add pairwise-distinctness or positive-radius hypotheses that are absent from the formal type.", "It does not assert the separate two-dimensional converse that a right angle among three points on an arbitrary sphere forces the first and third points to be diameter endpoints.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.thales-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Given 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.", "targetId": "target.library.mathlib.thales-theorem.v001", "title": "Thales’ Theorem", "upstreamOrigin": { "declarationName": "EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Geometry/Euclidean/Angle/Sphere.lean#L82", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:569d832b936006c5e4639cb2e83a0568d46722fdf3e1c6732635003594aa6950", "sourceByteLength": 31477, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Geometry/Euclidean/Angle/Sphere.lean", "sourceLine": 82, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }