Existing in Mathlib · Euclidean geometry

Ptolemy’s Theorem

Put four points on one sphere and order them so the two diagonals pass straight through a common point p. Then the product of one pair of opposite side lengths plus the product of the other pair equals the product of the diagonal lengths.

Ptolemy's theorem
cospherical points
Euclidean distance
cyclic quadrilaterals
crossing diagonals
inner-product spaces

Exact theorem

Exact Mathlib statement

theorem EuclideanGeometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {P : Type*} [MetricSpace P] [NormedAddTorsor V P] {a b c d p : P} (h : Cospherical ({a, b, c, d} : Set P)) (hapc : ∠ a p c = π) (hbpd : ∠ b p d = π) : dist a b * dist c d + dist b c * dist d a = dist a c * dist b d

The theorem at a glance

Ptolemy’s Theorem at a glance

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

An ivory Ptolemy theorem poster shows four cospherical points a, b, c, d, crossing diagonals through p, the two straight-angle hypotheses, and the exact sum-of-side-products equals diagonal-product identity. — For the cospherical order encoded by the two straight-angle hypotheses at p, the two opposite-side products add to the diagonal product.

Statement structure

Statement and scope

Read the exact Mathlib declaration

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

Four black-ringed points on one circular sphere section form a quadrilateral with cobalt opposite edges, emerald opposite edges, and two woven gold diagonals crossing at one interior point; three product tiles sit below. — Two visually paired opposite-side products combine into the product represented by the crossing diagonals.

Why it matters

A mathematical landmark

Ptolemy's theorem is one of classical Euclidean geometry's best-known metric identities, linking all six distances of a cyclic quadrilateral. Mathlib's formulation exposes the exact source-level ordering through a shared crossing point and works in a general real inner-product affine metric space.

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 cospherical four-point distance identity in a real inner-product affine metric space. It assumes an auxiliary point p with ∠ a p c = π and ∠ b p d = π, which encodes the crossing order through p, and concludes dist a b * dist c d + dist b c * dist d a = dist a c * dist b d. It does not separately assume an ordered planar polygon, convexity, pairwise-distinct vertices, or nondegeneracy, and it does not assert the equality for an arbitrary ordering of four cospherical points.

The selected declaration does not state Ptolemy's inequality or its converse.
The two angle-at-p hypotheses are essential to the selected ordering; the equality is not presented for an arbitrary ordering of four cospherical points.
The familiar cyclic-quadrilateral picture is a representative visualization, not an extra formal planar or convexity hypothesis.
The declaration adds no pairwise-distinctness or nondegeneracy hypothesis and makes no uniqueness, area, circumradius, or construction claim.
Proof Atlas is indexing an existing Mathlib theorem, and the generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

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