Documentation

Mathlib.Geometry.Euclidean.Circumcenter

Circumcenter and circumradius #

This file proves some lemmas on points equidistant from a set of points, and defines the circumradius and circumcenter of a simplex. There are also some definitions for use in calculations where it is convenient to work with affine combinations of vertices together with the circumcenter.

Main definitions #

References #

theorem EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} [Nonempty s] [HasOrthogonalProjection s.direction] {p1 : P} {p2 : P} (p3 : P) (hp1 : p1 s) (hp2 : p2 s) :

p is equidistant from two points in s if and only if its orthogonalProjection is.

theorem EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} [Nonempty s] [HasOrthogonalProjection s.direction] {ps : Set P} (hps : ps s) (p : P) :
(ps.Pairwise fun (p1 p2 : P) => dist p1 p = dist p2 p) ps.Pairwise fun (p1 p2 : P) => dist p1 ((EuclideanGeometry.orthogonalProjection s) p) = dist p2 ((EuclideanGeometry.orthogonalProjection s) p)

p is equidistant from a set of points in s if and only if its orthogonalProjection is.

theorem EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} [Nonempty s] [HasOrthogonalProjection s.direction] {ps : Set P} (hps : ps s) (p : P) :
(∃ (r : ), p1ps, dist p1 p = r) ∃ (r : ), p1ps, dist p1 ((EuclideanGeometry.orthogonalProjection s) p) = r

There exists r such that p has distance r from all the points of a set of points in s if and only if there exists (possibly different) r such that its orthogonalProjection has that distance from all the points in that set.

theorem EuclideanGeometry.existsUnique_dist_eq_of_insert {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} [HasOrthogonalProjection s.direction] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps s) (hp : ps) (hu : ∃! cs : EuclideanGeometry.Sphere P, cs.center s ps Metric.sphere cs.center cs.radius) :
∃! cs₂ : EuclideanGeometry.Sphere P, cs₂.center affineSpan (insert p s) insert p ps Metric.sphere cs₂.center cs₂.radius

The induction step for the existence and uniqueness of the circumcenter. Given a nonempty set of points in a nonempty affine subspace whose direction is complete, such that there is a unique (circumcenter, circumradius) pair for those points in that subspace, and a point p not in that subspace, there is a unique (circumcenter, circumradius) pair for the set with p added, in the span of the subspace with p added.

theorem AffineIndependent.existsUnique_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ιP} (ha : AffineIndependent p) :
∃! cs : EuclideanGeometry.Sphere P, cs.center affineSpan (Set.range p) Set.range p Metric.sphere cs.center cs.radius

Given a finite nonempty affinely independent family of points, there is a unique (circumcenter, circumradius) pair for those points in the affine subspace they span.

The circumsphere of a simplex.

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    theorem Affine.Simplex.circumsphere_unique_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) :
    (s.circumsphere.center affineSpan (Set.range s.points) Set.range s.points Metric.sphere s.circumsphere.center s.circumsphere.radius) ∀ (cs : EuclideanGeometry.Sphere P), cs.center affineSpan (Set.range s.points) Set.range s.points Metric.sphere cs.center cs.radiuscs = s.circumsphere

    The property satisfied by the circumsphere.

    The circumcenter of a simplex.

    Equations
    • s.circumcenter = s.circumsphere.center
    Instances For

      The circumradius of a simplex.

      Equations
      • s.circumradius = s.circumsphere.radius
      Instances For
        @[simp]
        theorem Affine.Simplex.circumsphere_center {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) :
        s.circumsphere.center = s.circumcenter

        The center of the circumsphere is the circumcenter.

        @[simp]
        theorem Affine.Simplex.circumsphere_radius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) :
        s.circumsphere.radius = s.circumradius

        The radius of the circumsphere is the circumradius.

        The circumcenter lies in the affine span.

        @[simp]
        theorem Affine.Simplex.dist_circumcenter_eq_circumradius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) (i : Fin (n + 1)) :
        dist (s.points i) s.circumcenter = s.circumradius

        All points have distance from the circumcenter equal to the circumradius.

        theorem Affine.Simplex.mem_circumsphere {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) (i : Fin (n + 1)) :
        s.points i s.circumsphere

        All points lie in the circumsphere.

        @[simp]
        theorem Affine.Simplex.dist_circumcenter_eq_circumradius' {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) (i : Fin (n + 1)) :
        dist s.circumcenter (s.points i) = s.circumradius

        All points have distance to the circumcenter equal to the circumradius.

        theorem Affine.Simplex.eq_circumcenter_of_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p : P} (hp : p affineSpan (Set.range s.points)) {r : } (hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r) :
        p = s.circumcenter

        Given a point in the affine span from which all the points are equidistant, that point is the circumcenter.

        theorem Affine.Simplex.eq_circumradius_of_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p : P} (hp : p affineSpan (Set.range s.points)) {r : } (hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r) :
        r = s.circumradius

        Given a point in the affine span from which all the points are equidistant, that distance is the circumradius.

        theorem Affine.Simplex.circumradius_nonneg {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) :
        0 s.circumradius

        The circumradius is non-negative.

        theorem Affine.Simplex.circumradius_pos {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P (n + 1)) :
        0 < s.circumradius

        The circumradius of a simplex with at least two points is positive.

        theorem Affine.Simplex.circumcenter_eq_point {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex P 0) (i : Fin 1) :
        s.circumcenter = s.points i

        The circumcenter of a 0-simplex equals its unique point.

        theorem Affine.Simplex.circumcenter_eq_centroid {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex P 1) :
        s.circumcenter = Finset.centroid Finset.univ s.points

        The circumcenter of a 1-simplex equals its centroid.

        @[simp]
        theorem Affine.Simplex.circumsphere_reindex {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {m : } {n : } (s : Affine.Simplex P m) (e : Fin (m + 1) Fin (n + 1)) :
        (s.reindex e).circumsphere = s.circumsphere

        Reindexing a simplex along an Equiv of index types does not change the circumsphere.

        @[simp]
        theorem Affine.Simplex.circumcenter_reindex {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {m : } {n : } (s : Affine.Simplex P m) (e : Fin (m + 1) Fin (n + 1)) :
        (s.reindex e).circumcenter = s.circumcenter

        Reindexing a simplex along an Equiv of index types does not change the circumcenter.

        @[simp]
        theorem Affine.Simplex.circumradius_reindex {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {m : } {n : } (s : Affine.Simplex P m) (e : Fin (m + 1) Fin (n + 1)) :
        (s.reindex e).circumradius = s.circumradius

        Reindexing a simplex along an Equiv of index types does not change the circumradius.

        The orthogonal projection of a point p onto the hyperplane spanned by the simplex's points.

        Equations
        Instances For
          theorem Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p1 : P} (p2 : P) (r : ) (hp : p1 affineSpan (Set.range s.points)) :
          s.orthogonalProjectionSpan (r (p2 -ᵥ (s.orthogonalProjectionSpan p2)) +ᵥ p1) = p1, hp

          Adding a vector to a point in the given subspace, then taking the orthogonal projection, produces the original point if the vector is a multiple of the result of subtracting a point's orthogonal projection from that point.

          theorem Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } {r₁ : } (s : Affine.Simplex P n) {p : P} {p₁o : P} (hp₁o : p₁o affineSpan (Set.range s.points)) :
          (s.orthogonalProjectionSpan (r₁ (p -ᵥ (s.orthogonalProjectionSpan p)) +ᵥ p₁o)) = p₁o
          theorem Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p1 : P} (p2 : P) (hp1 : p1 affineSpan (Set.range s.points)) :
          dist p1 p2 * dist p1 p2 = dist p1 (s.orthogonalProjectionSpan p2) * dist p1 (s.orthogonalProjectionSpan p2) + dist p2 (s.orthogonalProjectionSpan p2) * dist p2 (s.orthogonalProjectionSpan p2)
          theorem Affine.Simplex.dist_circumcenter_sq_eq_sq_sub_circumradius {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } {r : } (s : Affine.Simplex P n) {p₁ : P} (h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r) (h₁' : (s.orthogonalProjectionSpan p₁) = s.circumcenter) (h : s.points 0 affineSpan (Set.range s.points)) :
          dist p₁ s.circumcenter * dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius
          theorem Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p : P} (hr : ∃ (r : ), ∀ (i : Fin (n + 1)), dist (s.points i) p = r) :
          (s.orthogonalProjectionSpan p) = s.circumcenter

          If there exists a distance that a point has from all vertices of a simplex, the orthogonal projection of that point onto the subspace spanned by that simplex is its circumcenter.

          theorem Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {p : P} {r : } (hr : ∀ (i : Fin (n + 1)), dist (s.points i) p = r) :
          (s.orthogonalProjectionSpan p) = s.circumcenter

          If a point has the same distance from all vertices of a simplex, the orthogonal projection of that point onto the subspace spanned by that simplex is its circumcenter.

          theorem Affine.Simplex.orthogonalProjection_circumcenter {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {fs : Finset (Fin (n + 1))} {m : } (h : fs.card = m + 1) :
          ((s.face h).orthogonalProjectionSpan s.circumcenter) = (s.face h).circumcenter

          The orthogonal projection of the circumcenter onto a face is the circumcenter of that face.

          theorem Affine.Simplex.circumcenter_eq_of_range_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } {s₁ : Affine.Simplex P n} {s₂ : Affine.Simplex P n} (h : Set.range s₁.points = Set.range s₂.points) :
          s₁.circumcenter = s₂.circumcenter

          Two simplices with the same points have the same circumcenter.

          An index type for the vertices of a simplex plus its circumcenter. This is for use in calculations where it is convenient to work with affine combinations of vertices together with the circumcenter. (An equivalent form sometimes used in the literature is placing the circumcenter at the origin and working with vectors for the vertices.)

          Instances For
            Equations
            theorem Affine.Simplex.sum_pointsWithCircumcenter {α : Type u_3} [AddCommMonoid α] {n : } (f : Affine.Simplex.PointsWithCircumcenterIndex nα) :
            i : Affine.Simplex.PointsWithCircumcenterIndex n, f i = i : Fin (n + 1), f (Affine.Simplex.PointsWithCircumcenterIndex.pointIndex i) + f Affine.Simplex.PointsWithCircumcenterIndex.circumcenterIndex

            The sum of a function over PointsWithCircumcenterIndex.

            The vertices of a simplex plus its circumcenter.

            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
              @[simp]
              theorem Affine.Simplex.pointsWithCircumcenter_point {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) (i : Fin (n + 1)) :
              s.pointsWithCircumcenter (Affine.Simplex.PointsWithCircumcenterIndex.pointIndex i) = s.points i

              pointsWithCircumcenter, applied to a pointIndex value, equals points applied to that value.

              @[simp]
              theorem Affine.Simplex.pointsWithCircumcenter_eq_circumcenter {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) :
              s.pointsWithCircumcenter Affine.Simplex.PointsWithCircumcenterIndex.circumcenterIndex = s.circumcenter

              pointsWithCircumcenter, applied to circumcenterIndex, equals the circumcenter.

              The weights for a single vertex of a simplex, in terms of pointsWithCircumcenter.

              Equations
              • One or more equations did not get rendered due to their size.
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                @[simp]

                point_weights_with_circumcenter sums to 1.

                A single vertex, in terms of pointsWithCircumcenter.

                The weights for the centroid of some vertices of a simplex, in terms of pointsWithCircumcenter.

                Equations
                • One or more equations did not get rendered due to their size.
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                  The centroid of some vertices of a simplex, in terms of pointsWithCircumcenter.

                  The weights for the circumcenter of a simplex, in terms of pointsWithCircumcenter.

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For

                    The weights for the reflection of the circumcenter in an edge of a simplex. This definition is only valid with i₁ ≠ i₂.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      theorem Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } (s : Affine.Simplex P n) {i₁ : Fin (n + 1)} {i₂ : Fin (n + 1)} (h : i₁ i₂) :
                      (EuclideanGeometry.reflection (affineSpan (s.points '' {i₁, i₂}))) s.circumcenter = (Finset.affineCombination Finset.univ s.pointsWithCircumcenter) (Affine.Simplex.reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)

                      The reflection of the circumcenter of a simplex in an edge, in terms of pointsWithCircumcenter.

                      theorem EuclideanGeometry.cospherical_iff_exists_mem_of_complete {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] [HasOrthogonalProjection s.direction] :
                      EuclideanGeometry.Cospherical ps centers, ∃ (radius : ), pps, 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.

                      theorem EuclideanGeometry.cospherical_iff_exists_mem_of_finiteDimensional {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] [FiniteDimensional s.direction] :
                      EuclideanGeometry.Cospherical ps centers, ∃ (radius : ), pps, dist p center = radius

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

                      theorem EuclideanGeometry.exists_circumradius_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) :
                      ∃ (r : ), ∀ (sx : Affine.Simplex P n), Set.range sx.points pssx.circumradius = r

                      All n-simplices among cospherical points in an n-dimensional subspace have the same circumradius.

                      theorem EuclideanGeometry.circumradius_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumradius = sx₂.circumradius

                      Two n-simplices among cospherical points in an n-dimensional subspace have the same circumradius.

                      theorem EuclideanGeometry.exists_circumradius_eq_of_cospherical {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} {n : } [FiniteDimensional V] (hd : FiniteDimensional.finrank V = n) (hc : EuclideanGeometry.Cospherical ps) :
                      ∃ (r : ), ∀ (sx : Affine.Simplex P n), Set.range sx.points pssx.circumradius = r

                      All n-simplices among cospherical points in n-space have the same circumradius.

                      theorem EuclideanGeometry.circumradius_eq_of_cospherical {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} {n : } [FiniteDimensional V] (hd : FiniteDimensional.finrank V = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumradius = sx₂.circumradius

                      Two n-simplices among cospherical points in n-space have the same circumradius.

                      theorem EuclideanGeometry.exists_circumcenter_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) :
                      ∃ (c : P), ∀ (sx : Affine.Simplex P n), Set.range sx.points pssx.circumcenter = c

                      All n-simplices among cospherical points in an n-dimensional subspace have the same circumcenter.

                      theorem EuclideanGeometry.circumcenter_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumcenter = sx₂.circumcenter

                      Two n-simplices among cospherical points in an n-dimensional subspace have the same circumcenter.

                      theorem EuclideanGeometry.exists_circumcenter_eq_of_cospherical {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} {n : } [FiniteDimensional V] (hd : FiniteDimensional.finrank V = n) (hc : EuclideanGeometry.Cospherical ps) :
                      ∃ (c : P), ∀ (sx : Affine.Simplex P n), Set.range sx.points pssx.circumcenter = c

                      All n-simplices among cospherical points in n-space have the same circumcenter.

                      theorem EuclideanGeometry.circumcenter_eq_of_cospherical {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} {n : } [FiniteDimensional V] (hd : FiniteDimensional.finrank V = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumcenter = sx₂.circumcenter

                      Two n-simplices among cospherical points in n-space have the same circumcenter.

                      theorem EuclideanGeometry.exists_circumsphere_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) :
                      ∃ (c : EuclideanGeometry.Sphere P), ∀ (sx : Affine.Simplex P n), Set.range sx.points pssx.circumsphere = c

                      All n-simplices among cospherical points in an n-dimensional subspace have the same circumsphere.

                      theorem EuclideanGeometry.circumsphere_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {s : AffineSubspace P} {ps : Set P} (h : ps s) [Nonempty s] {n : } [FiniteDimensional s.direction] (hd : FiniteDimensional.finrank s.direction = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumsphere = sx₂.circumsphere

                      Two n-simplices among cospherical points in an n-dimensional subspace have the same circumsphere.

                      All n-simplices among cospherical points in n-space have the same circumsphere.

                      theorem EuclideanGeometry.circumsphere_eq_of_cospherical {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {ps : Set P} {n : } [FiniteDimensional V] (hd : FiniteDimensional.finrank V = n) (hc : EuclideanGeometry.Cospherical ps) {sx₁ : Affine.Simplex P n} {sx₂ : Affine.Simplex P n} (hsx₁ : Set.range sx₁.points ps) (hsx₂ : Set.range sx₂.points ps) :
                      sx₁.circumsphere = sx₂.circumsphere

                      Two n-simplices among cospherical points in n-space have the same circumsphere.

                      theorem EuclideanGeometry.eq_or_eq_reflection_of_dist_eq {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace V] [MetricSpace P] [NormedAddTorsor V P] {n : } {s : Affine.Simplex P n} {p : P} {p₁ : P} {p₂ : P} {r : } (hp₁ : p₁ affineSpan (insert p (Set.range s.points))) (hp₂ : p₂ affineSpan (insert p (Set.range s.points))) (h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r) (h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r) :
                      p₁ = p₂ p₁ = (EuclideanGeometry.reflection (affineSpan (Set.range s.points))) p₂

                      Suppose all distances from p₁ and p₂ to the points of a simplex are equal, and that p₁ and p₂ lie in the affine span of p with the vertices of that simplex. Then p₁ and p₂ are equal or reflections of each other in the affine span of the vertices of the simplex.