Theorems · Theorem · geometry

EuclideanGeometry.cospherical_iff_exists_mem_of_complete

∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] {s : AffineSubspace ℝ P} {ps : Set P},
  ps ⊆ ↑s →
    ∀ [Nonempty ↥s] [s.direction.HasOrthogonalProjection],
      EuclideanGeometry.Cospherical ps ↔ ∃ center ∈ s, ∃ radius, ∀ p ∈ ps, dist p center = radius

Given a nonempty affine subspace, whose direction is complete, that contains a set of points, those points are cospherical if and only if they are equidistant from some point in that subspace.

Defined in

Mathlib.Geometry.Euclidean.Circumcenter

Cited by

1 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection