mathlib3 documentation

geometry.euclidean.circumcenter

Circumcenter and circumradius #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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 #

source
theorem euclidean_geometry.dist_eq_iff_dist_orthogonal_projection_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} [nonempty ↥s] [complete_space ↥(s.direction)] {p1 p2 : P} (p3 : P) (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
has_dist.dist p1 p3 = has_dist.dist p2 p3 ↔ has_dist.dist p1 ↑(⇑(euclidean_geometry.orthogonal_projection s) p3) = has_dist.dist p2 ↑(⇑(euclidean_geometry.orthogonal_projection s) p3)

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

source
theorem euclidean_geometry.dist_set_eq_iff_dist_orthogonal_projection_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} [nonempty ↥s] [complete_space ↥(s.direction)] {ps : set P} (hps : ps ⊆ ↑s) (p : P) :
ps.pairwise (λ (p1 p2 : P), has_dist.dist p1 p = has_dist.dist p2 p) ↔ ps.pairwise (λ (p1 p2 : P), has_dist.dist p1 ↑(⇑(euclidean_geometry.orthogonal_projection s) p) = has_dist.dist p2 ↑(⇑(euclidean_geometry.orthogonal_projection s) p))

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

source
theorem euclidean_geometry.exists_dist_eq_iff_exists_dist_orthogonal_projection_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} [nonempty ↥s] [complete_space ↥(s.direction)] {ps : set P} (hps : ps ⊆ ↑s) (p : P) :
(∃ (r : ℝ), ∀ (p1 : P), p1 ∈ ps → has_dist.dist p1 p = r) ↔ ∃ (r : ℝ), ∀ (p1 : P), p1 ∈ ps → has_dist.dist p1 ↑(⇑(euclidean_geometry.orthogonal_projection 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 orthogonal_projection has that distance from all the points in that set.

source
theorem euclidean_geometry.exists_unique_dist_eq_of_insert {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} [complete_space ↥(s.direction)] {ps : set P} (hnps : ps.nonempty) {p : P} (hps : ps ⊆ ↑s) (hp : p ∉ s) (hu : ∃! (cs : euclidean_geometry.sphere P), cs.center ∈ s ∧ ps ⊆ ↑cs) :
∃! (cs₂ : euclidean_geometry.sphere P), cs₂.center ∈ affine_span ℝ (has_insert.insert p ↑s) ∧ has_insert.insert p ps ⊆ ↑cs₂

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.

source
theorem affine_independent.exists_unique_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ι : Type u_3} [hne : nonempty ι] [finite ι] {p : ι → P} (ha : affine_independent ℝ p) :
∃! (cs : euclidean_geometry.sphere P), cs.center ∈ affine_span ℝ (set.range p) ∧ set.range p ⊆ ↑cs

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.

source
noncomputable def affine.simplex.circumsphere {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
euclidean_geometry.sphere P

The circumsphere of a simplex.

Equations
source
theorem affine.simplex.circumsphere_unique_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
(s.circumsphere.center ∈ affine_span ℝ (set.range s.points) ∧ set.range s.points ⊆ ↑(s.circumsphere)) ∧ ∀ (cs : euclidean_geometry.sphere P), cs.center ∈ affine_span ℝ (set.range s.points) ∧ set.range s.points ⊆ ↑cs → cs = s.circumsphere

The property satisfied by the circumsphere.

source
noncomputable def affine.simplex.circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
P

The circumcenter of a simplex.

Equations
source
noncomputable def affine.simplex.circumradius {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
ℝ

The circumradius of a simplex.

Equations
source
@[simp]
theorem affine.simplex.circumsphere_center {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
s.circumsphere.center = s.circumcenter

The center of the circumsphere is the circumcenter.

source
@[simp]
theorem affine.simplex.circumsphere_radius {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
s.circumsphere.radius = s.circumradius

The radius of the circumsphere is the circumradius.

source
theorem affine.simplex.circumcenter_mem_affine_span {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
s.circumcenter ∈ affine_span ℝ (set.range s.points)

The circumcenter lies in the affine span.

source
@[simp]
theorem affine.simplex.dist_circumcenter_eq_circumradius {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (i : fin (n + 1)) :
has_dist.dist (s.points i) s.circumcenter = s.circumradius

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

source
theorem affine.simplex.mem_circumsphere {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (i : fin (n + 1)) :
s.points i ∈ s.circumsphere

All points lie in the circumsphere.

source
@[simp]
theorem affine.simplex.dist_circumcenter_eq_circumradius' {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (i : fin (n + 1)) :
has_dist.dist s.circumcenter (s.points i) = s.circumradius

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

source
theorem affine.simplex.eq_circumcenter_of_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p : P} (hp : p ∈ affine_span ℝ (set.range s.points)) {r : ℝ} (hr : ∀ (i : fin (n + 1)), has_dist.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.

source
theorem affine.simplex.eq_circumradius_of_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p : P} (hp : p ∈ affine_span ℝ (set.range s.points)) {r : ℝ} (hr : ∀ (i : fin (n + 1)), has_dist.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.

source
theorem affine.simplex.circumradius_nonneg {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
0 ≤ s.circumradius

The circumradius is non-negative.

source
theorem affine.simplex.circumradius_pos {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P (n + 1)) :
0 < s.circumradius

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

source
theorem affine.simplex.circumcenter_eq_point {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] (s : affine.simplex ℝ P 0) (i : fin 1) :
s.circumcenter = s.points i

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

source
theorem affine.simplex.circumcenter_eq_centroid {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] (s : affine.simplex ℝ P 1) :
s.circumcenter = finset.centroid ℝ finset.univ s.points

The circumcenter of a 1-simplex equals its centroid.

source
@[simp]
theorem affine.simplex.circumsphere_reindex {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {m n : ℕ} (s : affine.simplex ℝ 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.

source
@[simp]
theorem affine.simplex.circumcenter_reindex {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {m n : ℕ} (s : affine.simplex ℝ 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.

source
@[simp]
theorem affine.simplex.circumradius_reindex {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {m n : ℕ} (s : affine.simplex ℝ 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.

source
noncomputable def affine.simplex.orthogonal_projection_span {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
P →ᵃ[ℝ] ↥(affine_span ℝ (set.range s.points))

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

Equations
source
theorem affine.simplex.orthogonal_projection_vadd_smul_vsub_orthogonal_projection {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p1 : P} (p2 : P) (r : ℝ) (hp : p1 ∈ affine_span ℝ (set.range s.points)) :
⇑(s.orthogonal_projection_span) (r • (p2 -ᵥ ↑(⇑(s.orthogonal_projection_span) 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.

source
theorem affine.simplex.coe_orthogonal_projection_vadd_smul_vsub_orthogonal_projection {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} {r₁ : ℝ} (s : affine.simplex ℝ P n) {p p₁o : P} (hp₁o : p₁o ∈ affine_span ℝ (set.range s.points)) :
↑(⇑(s.orthogonal_projection_span) (r₁ • (p -ᵥ ↑(⇑(s.orthogonal_projection_span) p)) +ᵥ p₁o)) = p₁o
source
theorem affine.simplex.dist_sq_eq_dist_orthogonal_projection_sq_add_dist_orthogonal_projection_sq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p1 : P} (p2 : P) (hp1 : p1 ∈ affine_span ℝ (set.range s.points)) :
has_dist.dist p1 p2 * has_dist.dist p1 p2 = has_dist.dist p1 ↑(⇑(s.orthogonal_projection_span) p2) * has_dist.dist p1 ↑(⇑(s.orthogonal_projection_span) p2) + has_dist.dist p2 ↑(⇑(s.orthogonal_projection_span) p2) * has_dist.dist p2 ↑(⇑(s.orthogonal_projection_span) p2)
source
theorem affine.simplex.dist_circumcenter_sq_eq_sq_sub_circumradius {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} {r : ℝ} (s : affine.simplex ℝ P n) {p₁ : P} (h₁ : ∀ (i : fin (n + 1)), has_dist.dist (s.points i) p₁ = r) (h₁' : ↑(⇑(s.orthogonal_projection_span) p₁) = s.circumcenter) (h : s.points 0 ∈ affine_span ℝ (set.range s.points)) :
has_dist.dist p₁ s.circumcenter * has_dist.dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius
source
theorem affine.simplex.orthogonal_projection_eq_circumcenter_of_exists_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p : P} (hr : ∃ (r : ℝ), ∀ (i : fin (n + 1)), has_dist.dist (s.points i) p = r) :
↑(⇑(s.orthogonal_projection_span) 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.

source
theorem affine.simplex.orthogonal_projection_eq_circumcenter_of_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {p : P} {r : ℝ} (hr : ∀ (i : fin (n + 1)), has_dist.dist (s.points i) p = r) :
↑(⇑(s.orthogonal_projection_span) 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.

source
theorem affine.simplex.orthogonal_projection_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {fs : finset (fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) :
↑(⇑((s.face h).orthogonal_projection_span) s.circumcenter) = (s.face h).circumcenter

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

source
theorem affine.simplex.circumcenter_eq_of_range_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} {s₁ s₂ : affine.simplex ℝ 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.

source
@[protected, instance]
def affine.simplex.points_with_circumcenter_index.fintype (n : ℕ) :
fintype (affine.simplex.points_with_circumcenter_index n)
source
inductive affine.simplex.points_with_circumcenter_index (n : ℕ) :
Type

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 affine.simplex.points_with_circumcenter_index
source
@[protected, instance]
def affine.simplex.points_with_circumcenter_index_inhabited (n : ℕ) :
inhabited (affine.simplex.points_with_circumcenter_index n)
Equations
source
def affine.simplex.point_index_embedding (n : ℕ) :
fin (n + 1) ↪ affine.simplex.points_with_circumcenter_index n

point_index as an embedding.

Equations
source
theorem affine.simplex.sum_points_with_circumcenter {α : Type u_1} [add_comm_monoid α] {n : ℕ} (f : affine.simplex.points_with_circumcenter_index n → α) :
finset.univ.sum (λ (i : affine.simplex.points_with_circumcenter_index n), f i) = finset.univ.sum (λ (i : fin (n + 1)), f (affine.simplex.points_with_circumcenter_index.point_index i)) + f affine.simplex.points_with_circumcenter_index.circumcenter_index

The sum of a function over points_with_circumcenter_index.

source
noncomputable def affine.simplex.points_with_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
affine.simplex.points_with_circumcenter_index n → P

The vertices of a simplex plus its circumcenter.

Equations
source
@[simp]
theorem affine.simplex.points_with_circumcenter_point {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (i : fin (n + 1)) :
s.points_with_circumcenter (affine.simplex.points_with_circumcenter_index.point_index i) = s.points i

points_with_circumcenter, applied to a point_index value, equals points applied to that value.

source
@[simp]
theorem affine.simplex.points_with_circumcenter_eq_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
s.points_with_circumcenter affine.simplex.points_with_circumcenter_index.circumcenter_index = s.circumcenter

points_with_circumcenter, applied to circumcenter_index, equals the circumcenter.

source
def affine.simplex.point_weights_with_circumcenter {n : ℕ} (i : fin (n + 1)) :
affine.simplex.points_with_circumcenter_index n → ℝ

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

Equations
source
@[simp]
theorem affine.simplex.sum_point_weights_with_circumcenter {n : ℕ} (i : fin (n + 1)) :
finset.univ.sum (λ (j : affine.simplex.points_with_circumcenter_index n), affine.simplex.point_weights_with_circumcenter i j) = 1

point_weights_with_circumcenter sums to 1.

source
theorem affine.simplex.point_eq_affine_combination_of_points_with_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (i : fin (n + 1)) :
s.points i = ⇑(finset.affine_combination ℝ finset.univ s.points_with_circumcenter) (affine.simplex.point_weights_with_circumcenter i)

A single vertex, in terms of points_with_circumcenter.

source
noncomputable def affine.simplex.centroid_weights_with_circumcenter {n : ℕ} (fs : finset (fin (n + 1))) :
affine.simplex.points_with_circumcenter_index n → ℝ

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

Equations
source
@[simp]
theorem affine.simplex.sum_centroid_weights_with_circumcenter {n : ℕ} {fs : finset (fin (n + 1))} (h : fs.nonempty) :
finset.univ.sum (λ (i : affine.simplex.points_with_circumcenter_index n), affine.simplex.centroid_weights_with_circumcenter fs i) = 1

centroid_weights_with_circumcenter sums to 1, if the finset is nonempty.

source
theorem affine.simplex.centroid_eq_affine_combination_of_points_with_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) (fs : finset (fin (n + 1))) :
finset.centroid ℝ fs s.points = ⇑(finset.affine_combination ℝ finset.univ s.points_with_circumcenter) (affine.simplex.centroid_weights_with_circumcenter fs)

The centroid of some vertices of a simplex, in terms of points_with_circumcenter.

source
def affine.simplex.circumcenter_weights_with_circumcenter (n : ℕ) :
affine.simplex.points_with_circumcenter_index n → ℝ

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

Equations
source
@[simp]
theorem affine.simplex.sum_circumcenter_weights_with_circumcenter (n : ℕ) :
finset.univ.sum (λ (i : affine.simplex.points_with_circumcenter_index n), affine.simplex.circumcenter_weights_with_circumcenter n i) = 1

circumcenter_weights_with_circumcenter sums to 1.

source
theorem affine.simplex.circumcenter_eq_affine_combination_of_points_with_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) :
s.circumcenter = ⇑(finset.affine_combination ℝ finset.univ s.points_with_circumcenter) (affine.simplex.circumcenter_weights_with_circumcenter n)

The circumcenter of a simplex, in terms of points_with_circumcenter.

source
def affine.simplex.reflection_circumcenter_weights_with_circumcenter {n : ℕ} (i₁ i₂ : fin (n + 1)) :
affine.simplex.points_with_circumcenter_index n → ℝ

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

Equations
source
@[simp]
theorem affine.simplex.sum_reflection_circumcenter_weights_with_circumcenter {n : ℕ} {i₁ i₂ : fin (n + 1)} (h : i₁ ≠ i₂) :
finset.univ.sum (λ (i : affine.simplex.points_with_circumcenter_index n), affine.simplex.reflection_circumcenter_weights_with_circumcenter i₁ i₂ i) = 1

reflection_circumcenter_weights_with_circumcenter sums to 1.

source
theorem affine.simplex.reflection_circumcenter_eq_affine_combination_of_points_with_circumcenter {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} (s : affine.simplex ℝ P n) {i₁ i₂ : fin (n + 1)} (h : i₁ ≠ i₂) :
⇑(euclidean_geometry.reflection (affine_span ℝ (s.points '' {i₁, i₂}))) s.circumcenter = ⇑(finset.affine_combination ℝ finset.univ s.points_with_circumcenter) (affine.simplex.reflection_circumcenter_weights_with_circumcenter i₁ i₂)

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

source
theorem euclidean_geometry.cospherical_iff_exists_mem_of_complete {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] [complete_space ↥(s.direction)] :
euclidean_geometry.cospherical ps ↔ ∃ (center : P) (H : center ∈ s) (radius : ℝ), ∀ (p : P), p ∈ ps → has_dist.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.

source
theorem euclidean_geometry.cospherical_iff_exists_mem_of_finite_dimensional {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] [finite_dimensional ℝ ↥(s.direction)] :
euclidean_geometry.cospherical ps ↔ ∃ (center : P) (H : center ∈ s) (radius : ℝ), ∀ (p : P), p ∈ ps → has_dist.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.

source
theorem euclidean_geometry.exists_circumradius_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) :
∃ (r : ℝ), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumradius = r

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

source
theorem euclidean_geometry.circumradius_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.exists_circumradius_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) :
∃ (r : ℝ), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumradius = r

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

source
theorem euclidean_geometry.circumradius_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.exists_circumcenter_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) :
∃ (c : P), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumcenter = c

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

source
theorem euclidean_geometry.circumcenter_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.exists_circumcenter_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) :
∃ (c : P), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumcenter = c

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

source
theorem euclidean_geometry.circumcenter_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.exists_circumsphere_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) :
∃ (c : euclidean_geometry.sphere P), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumsphere = c

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

source
theorem euclidean_geometry.circumsphere_eq_of_cospherical_subset {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {s : affine_subspace ℝ P} {ps : set P} (h : ps ⊆ ↑s) [nonempty ↥s] {n : ℕ} [finite_dimensional ℝ ↥(s.direction)] (hd : fdim ↥(s.direction) = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.exists_circumsphere_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) :
∃ (c : euclidean_geometry.sphere P), ∀ (sx : affine.simplex ℝ P n), set.range sx.points ⊆ ps → sx.circumsphere = c

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

source
theorem euclidean_geometry.circumsphere_eq_of_cospherical {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {ps : set P} {n : ℕ} [finite_dimensional ℝ V] (hd : fdim V = n) (hc : euclidean_geometry.cospherical ps) {sx₁ sx₂ : affine.simplex ℝ 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.

source
theorem euclidean_geometry.eq_or_eq_reflection_of_dist_eq {V : Type u_1} {P : Type u_2} [normed_add_comm_group V] [inner_product_space ℝ V] [metric_space P] [normed_add_torsor V P] {n : ℕ} {s : affine.simplex ℝ P n} {p p₁ p₂ : P} {r : ℝ} (hp₁ : p₁ ∈ affine_span ℝ (has_insert.insert p (set.range s.points))) (hp₂ : p₂ ∈ affine_span ℝ (has_insert.insert p (set.range s.points))) (h₁ : ∀ (i : fin (n + 1)), has_dist.dist (s.points i) p₁ = r) (h₂ : ∀ (i : fin (n + 1)), has_dist.dist (s.points i) p₂ = r) :
p₁ = p₂ ∨ p₁ = ⇑(euclidean_geometry.reflection (affine_span ℝ (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.