Loogle!

Result

Found 342 declarations mentioning Affine.Simplex.points. Of these, only the first 200 are shown.

• Affine.Simplex.points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (self : Affine.Simplex k P n) : Fin (n + 1) → P
• Affine.Simplex.mkOfPoint_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  (k : Type u_1) {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) (i : Fin 1) : (Affine.Simplex.mkOfPoint k p).points i = p
• Affine.Simplex.independent 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (self : Affine.Simplex k P n) : AffineIndependent k self.points
• Affine.Simplex.range_mkOfPoint_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  (k : Type u_1) {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (p : P) : Set.range (Affine.Simplex.mkOfPoint k p).points = {p}
• Affine.Simplex.point_notMem_interior 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i ∉ s.interior
• Affine.Simplex.ext 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} {s1 s2 : Affine.Simplex k P n} (h : ∀ (i : Fin (n + 1)), s1.points i = s2.points i) : s1 = s2
• Affine.Simplex.ext_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} {s1 s2 : Affine.Simplex k P n} : s1 = s2 ↔ ∀ (i : Fin (n + 1)), s1.points i = s2.points i
• Affine.Simplex.point_mem_closedInterior 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [ZeroLEOneClass k] {n : ℕ} (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i ∈ s.closedInterior
• Affine.Simplex.closedInterior_subset_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} {s : Affine.Simplex k P n} : s.closedInterior ⊆ ↑(affineSpan k (Set.range s.points))
• Affine.Simplex.setInterior_subset_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {I : Set k} {n : ℕ} {s : Affine.Simplex k P n} : Affine.Simplex.setInterior I s ⊆ ↑(affineSpan k (Set.range s.points))
• Affine.Simplex.faceOpposite_point_eq_point_rev 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Affine.Simplex k P 1) (i : Fin 2) (n : Fin 1) : (s.faceOpposite i).points n = s.points i.rev
• Affine.Simplex.reindex_range_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} (s : Affine.Simplex k P m) (e : Fin (m + 1) ≃ Fin (n + 1)) : Set.range (s.reindex e).points = Set.range s.points
• Affine.Simplex.closedInterior_eq_singleton 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [ZeroLEOneClass k] (s : Affine.Simplex k P 0) : s.closedInterior = {s.points 0}
• Affine.Simplex.face_eq_mkOfPoint 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.face ⋯ = Affine.Simplex.mkOfPoint k (s.points i)
• Affine.Simplex.point_mem_closedInterior_faceOpposite_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [Nontrivial k] [ZeroLEOneClass k] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) {i j : Fin (n + 1)} : s.points j ∈ (s.faceOpposite i).closedInterior ↔ j ≠ i
• Affine.Simplex.points_notMem_affineSpan_faceOpposite 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Nontrivial k] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i ∉ affineSpan k (Set.range (s.faceOpposite i).points)
• Affine.Simplex.faceOpposite_point_eq_point_succAbove 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) (j : Fin (n - 1 + 1)) : (s.faceOpposite i).points j = s.points (i.succAbove (Fin.cast ⋯ j))
• Affine.Simplex.mem_affineSpan_image_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Nontrivial k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Set (Fin (n + 1))} {i : Fin (n + 1)} : s.points i ∈ affineSpan k (s.points '' fs) ↔ i ∈ fs
• Affine.Simplex.affineSpan_faceOpposite_le 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : affineSpan k (Set.range (s.faceOpposite i).points) ≤ affineSpan k (Set.range s.points)
• Affine.Simplex.range_face_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : Set.range (s.face h).points = s.points '' ↑fs
• Affine.Simplex.affineSpan_face_le 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : affineSpan k (Set.range (s.face h).points) ≤ affineSpan k (Set.range s.points)
• Affine.Simplex.points_mem_affineSpan_faceOpposite 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Nontrivial k] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) {i j : Fin (n + 1)} : s.points j ∈ affineSpan k (Set.range (s.faceOpposite i).points) ↔ j ≠ i
• Affine.Simplex.map_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {V₂ : Type u_3} {P : Type u_5} {P₂ : Type u_6} [Ring k] [AddCommGroup V] [AddCommGroup V₂] [Module k V] [Module k V₂] [AddTorsor V P] [AddTorsor V₂ P₂] {n : ℕ} (s : Affine.Simplex k P n) (f : P →ᵃ[k] P₂) (hf : Function.Injective ⇑f) : (s.map f hf).points = ⇑f ∘ s.points
• Affine.Simplex.faceOpposite_point_eq_point_one 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Affine.Simplex k P 1) (n : Fin 1) : (s.faceOpposite 0).points n = s.points 1
• Affine.Simplex.faceOpposite_point_eq_point_zero 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (s : Affine.Simplex k P 1) (n : Fin 1) : (s.faceOpposite 1).points n = s.points 0
• Affine.Simplex.range_faceOpposite_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : Set.range (s.faceOpposite i).points = s.points '' {i}ᶜ
• Affine.Simplex.point_mem_closedInterior_face_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [Nontrivial k] [ZeroLEOneClass k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {j : Fin (n + 1)} : s.points j ∈ (s.face h).closedInterior ↔ j ∈ fs
• Affine.Simplex.points_mem_affineSpan_face 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [Nontrivial k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {i : Fin (n + 1)} : s.points i ∈ affineSpan k (Set.range (s.face h).points) ↔ i ∈ fs
• Affine.Simplex.reindex_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} (s : Affine.Simplex k P m) (e : Fin (m + 1) ≃ Fin (n + 1)) (a✝ : Fin (n + 1)) : (s.reindex e).points a✝ = (s.points ∘ ⇑e.symm) a✝
• Affine.Simplex.range_faceOpposite_reindex 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} [NeZero m] [NeZero n] (s : Affine.Simplex k P m) (e : Fin (m + 1) ≃ Fin (n + 1)) (i : Fin (n + 1)) : Set.range ((s.reindex e).faceOpposite i).points = Set.range (s.faceOpposite (e.symm i)).points
• Affine.Simplex.face_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) (i : Fin (m + 1)) : (s.face h).points i = s.points ((fs.orderEmbOfFin h) i)
• Affine.Simplex.affineCombination_mem_setInterior_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {I : Set k} {n : ℕ} {s : Affine.Simplex k P n} {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ Affine.Simplex.setInterior I s ↔ ∀ (i : Fin (n + 1)), w i ∈ I
• Affine.Simplex.range_face_reindex 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} (s : Affine.Simplex k P m) (e : Fin (m + 1) ≃ Fin (n + 1)) {fs : Finset (Fin (n + 1))} {n' : ℕ} (h : fs.card = n' + 1) : Set.range ((s.reindex e).face h).points = Set.range (s.face ⋯).points
• Affine.Simplex.face_points' 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : (s.face h).points = s.points ∘ ⇑(fs.orderEmbOfFin h)
• Affine.Simplex.affineCombination_mem_closedInterior_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} {s : Affine.Simplex k P n} {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ s.closedInterior ↔ ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
• Affine.Simplex.affineCombination_mem_interior_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} {s : Affine.Simplex k P n} {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ s.interior ↔ ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
• Affine.Simplex.affineCombination_mem_affineSpan_faceOpposite_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex k P n} {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) {i : Fin (n + 1)} : (Finset.affineCombination k Finset.univ s.points) w ∈ affineSpan k (Set.range (s.faceOpposite i).points) ↔ w i = 0
• Affine.Simplex.affineCombination_mem_setInterior_face_iff_mem 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (I : Set k) {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ Affine.Simplex.setInterior I (s.face h) ↔ (∀ i ∈ fs, w i ∈ I) ∧ ∀ i ∉ fs, w i = 0
• Affine.Simplex.affineCombination_mem_closedInterior_face_iff_nonneg 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [IsOrderedAddMonoid k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ (s.face h).closedInterior ↔ (∀ i ∈ fs, 0 ≤ w i) ∧ ∀ i ∉ fs, w i = 0
• Affine.Simplex.affineCombination_mem_closedInterior_face_iff_mem_Icc 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ (s.face h).closedInterior ↔ (∀ i ∈ fs, w i ∈ Set.Icc 0 1) ∧ ∀ i ∉ fs, w i = 0
• Affine.Simplex.affineCombination_mem_interior_face_iff_mem_Ioo 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ (s.face h).interior ↔ (∀ i ∈ fs, w i ∈ Set.Ioo 0 1) ∧ ∀ i ∉ fs, w i = 0
• Affine.Simplex.affineCombination_mem_interior_face_iff_pos 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] [IsOrderedAddMonoid k] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} [NeZero m] (h : fs.card = m + 1) {w : Fin (n + 1) → k} (hw : ∑ i, w i = 1) : (Finset.affineCombination k Finset.univ s.points) w ∈ (s.face h).interior ↔ (∀ i ∈ fs, 0 < w i) ∧ ∀ i ∉ fs, w i = 0
• Affine.Simplex.restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : Affine.Simplex k (↥S) n
• Affine.Simplex.restrict_points_coe 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) (i : Fin (n + 1)) : ↑((s.restrict S hS).points i) = s.points i
• Affine.Simplex.face_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : (s.restrict S hS).face h = (s.face h).restrict S ⋯
• Affine.Simplex.faceOpposite_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) (i : Fin (n + 1)) : (s.restrict S hS).faceOpposite i = (s.faceOpposite i).restrict S ⋯
• Affine.Simplex.restrict_reindex 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {m n : ℕ} (s : Affine.Simplex k P n) (e : Fin (n + 1) ≃ Fin (m + 1)) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.reindex e).restrict S ⋯ = (s.restrict S hS).reindex e
• Affine.Simplex.restrict_map_subtype 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) : (s.restrict (affineSpan k (Set.range s.points)) ⋯).map (affineSpan k (Set.range s.points)).subtype ⋯ = s
• Affine.Simplex.closedInterior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).closedInterior = ⇑S.subtype ⁻¹' s.closedInterior
• Affine.Simplex.interior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] [PartialOrder k] {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).interior = ⇑S.subtype ⁻¹' s.interior
• Affine.Simplex.setInterior_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_4} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] (I : Set k) {n : ℕ} (s : Affine.Simplex k P n) {S : AffineSubspace k P} (hS : affineSpan k (Set.range s.points) ≤ S) : Affine.Simplex.setInterior I (s.restrict S hS) = ⇑S.subtype ⁻¹' Affine.Simplex.setInterior I s
• Affine.Simplex.restrict_map_inclusion 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) (S₁ S₂ : AffineSubspace k P) (hS₁ : affineSpan k (Set.range s.points) ≤ S₁) (hS₂ : S₁ ≤ S₂) : (s.restrict S₁ hS₁).map (AffineSubspace.inclusion hS₂) ⋯ = s.restrict S₂ ⋯
• Affine.Simplex.map_subtype_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {P : Type u_5} [Ring k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (S : AffineSubspace k P) [Nonempty ↥S] (s : Affine.Simplex k (↥S) n) : (s.map S.subtype ⋯).restrict S ⋯ = s
• Affine.Simplex.restrict_map_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
  {k : Type u_1} {V : Type u_2} {V₂ : Type u_3} {P : Type u_5} {P₂ : Type u_6} [Ring k] [AddCommGroup V] [AddCommGroup V₂] [Module k V] [Module k V₂] [AddTorsor V P] [AddTorsor V₂ P₂] {n : ℕ} (s : Affine.Simplex k P n) (f : P →ᵃ[k] P₂) (hf : Function.Injective ⇑f) (S₁ : AffineSubspace k P) (S₂ : AffineSubspace k P₂) (hS₁ : affineSpan k (Set.range s.points) ≤ S₁) (hfS : AffineSubspace.map f S₁ ≤ S₂) : (s.restrict S₁ hS₁).map (f.restrict hfS) ⋯ = (s.map f hf).restrict S₂ ⋯
• Affine.Simplex.convexHull_eq_closedInterior 📋 Mathlib.Analysis.Convex.Combination
  {𝕜 : Type u_1} {V : Type u_2} [Field 𝕜] [LinearOrder 𝕜] [IsOrderedRing 𝕜] [AddCommGroup V] [Module 𝕜 V] {n : ℕ} (s : Affine.Simplex 𝕜 V n) : (convexHull 𝕜) (Set.range s.points) = s.closedInterior
• Affine.Simplex.medial_points 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.medial.points i = s.faceOppositeCentroid i
• Affine.Simplex.univ_centroid_eq 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) : Finset.centroid k Finset.univ s.points = s.centroid
• Affine.Simplex.point_mem_median 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i ∈ s.median i
• Affine.Simplex.centroid_mem_affineSpan 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} (s : Affine.Simplex k P n) : s.centroid ∈ affineSpan k (Set.range s.points)
• Affine.Simplex.median_eq_line_point_centroid 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.median i = line[k, s.points i, s.centroid]
• Affine.Simplex.affineSpan_range_medial 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) : affineSpan k (Set.range s.medial.points) = affineSpan k (Set.range s.points)
• Affine.Simplex.affineIndependent_points_update_centroid 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : AffineIndependent k (Function.update s.points i s.centroid)
• Affine.Simplex.centroid_weighted_vsub_eq_zero 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) : ∑ i, (s.points i -ᵥ s.centroid) = 0
• Affine.Simplex.eq_centroid_iff_sum_vsub_eq_zero 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] {s : Affine.Simplex k P n} {p : P} : p = s.centroid ↔ ∑ i, (s.points i -ᵥ p) = 0
• Affine.Simplex.centroid_notMem_affineSpan_of_ne_univ 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) {t : Set (Fin (n + 1))} (ht : t ≠ Set.univ) : s.centroid ∉ affineSpan k (s.points '' t)
• Affine.Simplex.faceOppositeCentroid_mem_affineSpan_face 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i ∈ affineSpan k (Set.range (s.faceOpposite i).points)
• Affine.Simplex.face_centroid_eq_centroid 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (h : fs.card = m + 1) : Finset.centroid k Finset.univ (s.face h).points = Finset.centroid k fs s.points
• Affine.Simplex.centroid_vsub_point_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.centroid -ᵥ s.points i = ↑n • (s.faceOppositeCentroid i -ᵥ s.centroid)
• Affine.Simplex.point_vsub_centroid_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i -ᵥ s.centroid = ↑n • (s.centroid -ᵥ s.faceOppositeCentroid i)
• Affine.Simplex.centroid_vsub_faceOppositeCentroid_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.centroid -ᵥ s.faceOppositeCentroid i = (↑n)⁻¹ • (s.points i -ᵥ s.centroid)
• Affine.Simplex.faceOppositeCentroid_vsub_centroid_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i -ᵥ s.centroid = (↑n)⁻¹ • (s.centroid -ᵥ s.points i)
• Affine.Simplex.centroid_eq_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} (s : Affine.Simplex k P n) {fs₁ fs₂ : Finset (Fin (n + 1))} {m₁ m₂ : ℕ} (h₁ : fs₁.card = m₁ + 1) (h₂ : fs₂.card = m₂ + 1) : Finset.centroid k fs₁ s.points = Finset.centroid k fs₂ s.points ↔ fs₁ = fs₂
• Affine.Simplex.centroid_eq_of_range_eq 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} {s₁ s₂ : Affine.Simplex k P n} (h : Set.range s₁.points = Set.range s₂.points) : Finset.centroid k Finset.univ s₁.points = Finset.centroid k Finset.univ s₂.points
• Affine.Simplex.centroid_eq_smul_vsub_vadd_point 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.centroid = ↑n • (s.faceOppositeCentroid i -ᵥ s.centroid) +ᵥ s.points i
• Affine.Simplex.faceOppositeCentroid_vsub_point_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i -ᵥ s.points i = (↑n + 1) • (s.faceOppositeCentroid i -ᵥ s.centroid)
• Affine.Simplex.point_vsub_faceOppositeCentroid_eq_smul_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i -ᵥ s.faceOppositeCentroid i = (↑n + 1) • (s.centroid -ᵥ s.faceOppositeCentroid i)
• Affine.Simplex.faceOppositeCentroid_vsub_faceOppositeCentroid 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i j : Fin (n + 1)) : s.faceOppositeCentroid i -ᵥ s.faceOppositeCentroid j = (↑n)⁻¹ • (s.points j -ᵥ s.points i)
• Affine.Simplex.faceOppositeCentroid_eq_smul_vsub_vadd_point 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i = (↑n)⁻¹ • (s.centroid -ᵥ s.points i) +ᵥ s.centroid
• Affine.Simplex.centroid_vsub_eq 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) (p : P) : s.centroid -ᵥ p = (↑n + 1)⁻¹ • ∑ x, (s.points x -ᵥ p)
• Affine.Simplex.smul_faceOppositeCentroid_vsub_point_eq_sum_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : ↑n • (s.faceOppositeCentroid i -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points i)
• Affine.Simplex.faceOppositeCentroid_vsub_point_eq_smul_sum_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i -ᵥ s.points i = (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i)
• Affine.Simplex.point_vsub_faceOppositeCentroid_eq_smul_sum_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.points i -ᵥ s.faceOppositeCentroid i = (↑n)⁻¹ • ∑ x, (s.points i -ᵥ s.points x)
• Affine.Simplex.smul_centroid_vsub_point_eq_sum_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : (↑n + 1) • (s.centroid -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points i)
• Affine.Simplex.face_centroid_eq_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} (s : Affine.Simplex k P n) {fs₁ fs₂ : Finset (Fin (n + 1))} {m₁ m₂ : ℕ} (h₁ : fs₁.card = m₁ + 1) (h₂ : fs₂.card = m₂ + 1) : Finset.centroid k Finset.univ (s.face h₁).points = Finset.centroid k Finset.univ (s.face h₂).points ↔ fs₁ = fs₂
• Affine.Simplex.smul_centroid_vsub_point_eq_smul_faceOppositeCentroid_vsub_point 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : (↑n + 1) • (s.centroid -ᵥ s.points i) = ↑n • (s.faceOppositeCentroid i -ᵥ s.points i)
• Affine.Simplex.faceOppositeCentroid_eq_sum_vsub_vadd 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i = (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) +ᵥ s.points i
• Affine.Simplex.centroid_eq_smul_sum_vsub_vadd 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.centroid = (↑n + 1)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) +ᵥ s.points i
• Affine.Simplex.centroid_eq_affineCombination 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} (s : Affine.Simplex k P n) : s.centroid = (Finset.affineCombination k Finset.univ s.points) (Finset.centroidWeights k Finset.univ)
• Affine.Simplex.faceOppositeCentroid_eq_affineCombination 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : s.faceOppositeCentroid i = (Finset.affineCombination k {i}ᶜ s.points) fun x => (↑n)⁻¹
• Affine.Simplex.centroid_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : ↑(s.restrict S hS).centroid = s.centroid
• Affine.Simplex.faceOppositeCentroid_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) {i : Fin (n + 1)} : ↑((s.restrict S hS).faceOppositeCentroid i) = s.faceOppositeCentroid i
• Affine.Simplex.medial_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : (s.restrict S hS).medial = s.medial.restrict S ⋯
• Affine.Simplex.median_restrict 📋 Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)) (S : AffineSubspace k P) (hS : affineSpan k (Set.range s.points) ≤ S) : AffineSubspace.map S.subtype ((s.restrict S hS).median i) = s.median i
• Affine.Simplex.collinear_point_centroid_faceOppositeCentroid 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] [CharZero k] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) : Collinear k {s.points i, s.centroid, s.faceOppositeCentroid i}
• Affine.Simplex.fact_finrank_direction_affineSpan_eq 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} {s : Affine.Simplex k P n} : Fact (Module.finrank k ↥(affineSpan k (Set.range s.points)).direction = n)
• Affine.Simplex.span_eq_top 📋 Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
  {k : Type u_1} {V : Type u_2} [DivisionRing k] [AddCommGroup V] [Module k V] [FiniteDimensional k V] {n : ℕ} (T : Affine.Simplex k V n) (hrank : Module.finrank k V = n) : affineSpan k (Set.range T.points) = ⊤
• Affine.Simplex.closedInterior_eq_affineSegment 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] (s : Affine.Simplex R P 1) : s.closedInterior = affineSegment R (s.points 0) (s.points 1)
• Affine.Simplex.mem_closedInterior_iff_wbtw 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] {s : Affine.Simplex R P 1} {p : P} : p ∈ s.closedInterior ↔ Wbtw R (s.points 0) p (s.points 1)
• Affine.Simplex.mem_interior_iff_sbtw 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] [IsDomain R] [Module.IsTorsionFree R V] {s : Affine.Simplex R P 1} {p : P} : p ∈ s.interior ↔ Sbtw R (s.points 0) p (s.points 1)
• Affine.Simplex.closedInterior_face_eq_affineSegment 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] {n : ℕ} (s : Affine.Simplex R P n) {i j : Fin (n + 1)} (h : i ≠ j) : (s.face ⋯).closedInterior = affineSegment R (s.points i) (s.points j)
• Affine.Simplex.mem_closedInterior_face_iff_wbtw 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] {n : ℕ} (s : Affine.Simplex R P n) {p : P} {i j : Fin (n + 1)} (h : i ≠ j) : p ∈ (s.face ⋯).closedInterior ↔ Wbtw R (s.points i) p (s.points j)
• Affine.Simplex.interior_eq_image_Ioo 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] (s : Affine.Simplex R P 1) : s.interior = ⇑(AffineMap.lineMap (s.points 0) (s.points 1)) '' Set.Ioo 0 1
• Affine.Simplex.mem_interior_face_iff_sbtw 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [PartialOrder R] [AddCommGroup V] [Module R V] [AddTorsor V P] [IsOrderedRing R] [IsDomain R] [Module.IsTorsionFree R V] {n : ℕ} (s : Affine.Simplex R P n) {p : P} {i j : Fin (n + 1)} (h : i ≠ j) : p ∈ (s.face ⋯).interior ↔ Sbtw R (s.points i) p (s.points j)
• sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair 📋 Mathlib.Analysis.Convex.Between
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] [Module.IsTorsionFree R V] {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P} (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)) (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) : Sbtw R (t.points i₁) p p₁
• Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide (s.points i) ((Finset.affineCombination R Finset.univ s.points) w) ↔ w i < 0
• Affine.Simplex.sOppSide_affineSpan_faceOpposite_point_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide ((Finset.affineCombination R Finset.univ s.points) w) (s.points i) ↔ w i < 0
• Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide (s.points i) ((Finset.affineCombination R Finset.univ s.points) w) ↔ 0 < w i
• Affine.Simplex.sSameSide_affineSpan_faceOpposite_point_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w) (s.points i) ↔ 0 < w i
• Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide (s.points i) ((Finset.affineCombination R Finset.univ s.points) w) ↔ w i ≤ 0
• Affine.Simplex.wOppSide_affineSpan_faceOpposite_point_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w) (s.points i) ↔ w i ≤ 0
• Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_left_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide (s.points i) ((Finset.affineCombination R Finset.univ s.points) w) ↔ 0 ≤ w i
• Affine.Simplex.wSameSide_affineSpan_faceOpposite_point_right_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w : Fin (n + 1) → R} (hw : ∑ j, w j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide ((Finset.affineCombination R Finset.univ s.points) w) (s.points i) ↔ 0 ≤ w i
• Affine.Simplex.sOppSide_affineSpan_faceOpposite_of_pos_of_neg 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} (hs₁ : 0 < w₁ i) (hs₂ : w₂ i < 0) : (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂)
• Affine.Simplex.sSameSide_affineSpan_faceOpposite_of_sign_eq 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} (hs : SignType.sign (w₁ i) = SignType.sign (w₂ i)) (h0 : w₁ i ≠ 0) : (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂)
• Affine.Simplex.sSameSide_affineSpan_faceOpposite_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SSameSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂) ↔ SignType.sign (w₁ i) = SignType.sign (w₂ i) ∧ w₁ i ≠ 0
• Affine.Simplex.sOppSide_affineSpan_faceOpposite_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).SOppSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂) ↔ SignType.sign (w₁ i) = -SignType.sign (w₂ i) ∧ w₁ i ≠ 0
• Affine.Simplex.wSameSide_affineSpan_faceOpposite_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂) ↔ SignType.sign (w₁ i) = SignType.sign (w₂ i) ∨ w₁ i = 0 ∨ w₂ i = 0
• Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff 📋 Mathlib.Analysis.Convex.Side
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Field R] [LinearOrder R] [IsStrictOrderedRing R] [AddCommGroup V] [Module R V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R} (hw₁ : ∑ j, w₁ j = 1) (hw₂ : ∑ j, w₂ j = 1) {i : Fin (n + 1)} : (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁) ((Finset.affineCombination R Finset.univ s.points) w₂) ↔ SignType.sign (w₁ i) = -SignType.sign (w₂ i) ∨ w₁ i = 0 ∨ w₂ i = 0
• Affine.Simplex.dist_lt_of_mem_interior_of_strictConvexSpace 📋 Mathlib.Analysis.Convex.StrictCombination
  {V : Type u_2} {P : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) {r : ℝ} {p₀ p : P} (hp : p ∈ s.interior) (hr : ∀ (i : Fin (n + 1)), dist (s.points i) p₀ ≤ r) : dist p p₀ < r
• Affine.Simplex.dist_lt_of_mem_closedInterior_of_strictConvexSpace 📋 Mathlib.Analysis.Convex.StrictCombination
  {V : Type u_2} {P : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [StrictConvexSpace ℝ V] [PseudoMetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) {r : ℝ} {p₀ p : P} (hp : p ∈ s.closedInterior) (hp' : ∀ (i : Fin (n + 1)), p ≠ s.points i) (hr : ∀ (i : Fin (n + 1)), dist (s.points i) p₀ ≤ r) : dist p p₀ < r
• Affine.Triangle.prod_eq_prod_one_sub_of_mem_line_point_lineMap 📋 Mathlib.LinearAlgebra.AffineSpace.Ceva
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [CommRing k] [NoZeroDivisors k] [AddCommGroup V] [Module k V] [AddTorsor V P] {t : Affine.Triangle k P} {r : Fin 3 → k} {p' : P} (hp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]) : ∏ i, r i = ∏ i, (1 - r i)
• Affine.Triangle.prod_div_one_sub_eq_one_of_mem_line_point_lineMap 📋 Mathlib.LinearAlgebra.AffineSpace.Ceva
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [AddCommGroup V] [Module k V] [AddTorsor V P] {t : Affine.Triangle k P} {r : Fin 3 → k} (hr0 : ∀ (i : Fin 3), r i ≠ 0) {p' : P} (hp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]) : ∏ i, r i / (1 - r i) = 1
• Affine.Triangle.prod_dist_eq_prod_dist_of_mem_line_of_mem_line 📋 Mathlib.Analysis.Normed.Affine.Ceva
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [NormedField 𝕜] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] {t : Affine.Triangle 𝕜 P} {p : Fin 3 → P} {p' : P} (hp : ∀ (i : Fin 3), p i ∈ line[𝕜, t.points (i + 1), t.points (i + 2)]) (hp' : ∀ (i : Fin 3), p' ∈ line[𝕜, t.points i, p i]) : ∏ i, dist (t.points (i + 1)) (p i) = ∏ i, dist (p i) (t.points (i + 2))
• Affine.Triangle.prod_dist_div_dist_eq_one_of_mem_line_of_mem_line 📋 Mathlib.Analysis.Normed.Affine.Ceva
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [NormedField 𝕜] [NormedSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {t : Affine.Triangle 𝕜 P} {p : Fin 3 → P} {p' : P} (hp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)) (hp : ∀ (i : Fin 3), p i ∈ line[𝕜, t.points (i + 1), t.points (i + 2)]) (hp' : ∀ (i : Fin 3), p' ∈ line[𝕜, t.points i, p i]) : ∏ i, dist (t.points (i + 1)) (p i) / dist (p i) (t.points (i + 2)) = 1
• Affine.Simplex.Equilateral.dist_eq 📋 Mathlib.Analysis.Normed.Affine.Simplex
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [SeminormedAddCommGroup V] [PseudoMetricSpace P] [Module R V] [NormedAddTorsor V P] {n : ℕ} {s : Affine.Simplex R P n} (he : s.Equilateral) {i₁ i₂ i₃ i₄ : Fin (n + 1)} (h₁₂ : i₁ ≠ i₂) (h₃₄ : i₃ ≠ i₄) : dist (s.points i₁) (s.points i₂) = dist (s.points i₃) (s.points i₄)
• Affine.Simplex.Scalene.dist_ne 📋 Mathlib.Analysis.Normed.Affine.Simplex
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [SeminormedAddCommGroup V] [PseudoMetricSpace P] [Module R V] [NormedAddTorsor V P] {n : ℕ} {s : Affine.Simplex R P n} (hs : s.Scalene) {i₁ i₂ i₃ i₄ : Fin (n + 1)} (h₁₂ : i₁ ≠ i₂) (h₃₄ : i₃ ≠ i₄) (h₁₂₃₄ : ¬(i₁ = i₃ ∧ i₂ = i₄)) (h₁₂₄₃ : ¬(i₁ = i₄ ∧ i₂ = i₃)) : dist (s.points i₁) (s.points i₂) ≠ dist (s.points i₃) (s.points i₄)
• Affine.Triangle.equilateral_iff_dist_eq_and_dist_eq 📋 Mathlib.Analysis.Normed.Affine.Simplex
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [SeminormedAddCommGroup V] [PseudoMetricSpace P] [Module R V] [NormedAddTorsor V P] {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : Affine.Simplex.Equilateral t ↔ dist (t.points i₁) (t.points i₂) = dist (t.points i₁) (t.points i₃) ∧ dist (t.points i₁) (t.points i₂) = dist (t.points i₂) (t.points i₃)
• Affine.Triangle.equilateral_iff_dist_01_eq_02_and_dist_01_eq_12 📋 Mathlib.Analysis.Normed.Affine.Simplex
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [SeminormedAddCommGroup V] [PseudoMetricSpace P] [Module R V] [NormedAddTorsor V P] {t : Affine.Triangle R P} : Affine.Simplex.Equilateral t ↔ dist (t.points 0) (t.points 1) = dist (t.points 0) (t.points 2) ∧ dist (t.points 0) (t.points 1) = dist (t.points 1) (t.points 2)
• Affine.Triangle.scalene_iff_dist_ne_and_dist_ne_and_dist_ne 📋 Mathlib.Analysis.Normed.Affine.Simplex
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [Ring R] [SeminormedAddCommGroup V] [PseudoMetricSpace P] [Module R V] [NormedAddTorsor V P] {t : Affine.Triangle R P} : Affine.Simplex.Scalene t ↔ dist (t.points 0) (t.points 1) ≠ dist (t.points 0) (t.points 2) ∧ dist (t.points 0) (t.points 1) ≠ dist (t.points 1) (t.points 2) ∧ dist (t.points 0) (t.points 2) ≠ dist (t.points 1) (t.points 2)
• Affine.Simplex.orthogonalProjectionSpan 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) : P →ᴬ[𝕜] ↥(affineSpan 𝕜 (Set.range s.points))
• Affine.Simplex.orthogonalProjectionSpan_eq_point 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex 𝕜 P 0) (p : P) : ↑(s.orthogonalProjectionSpan p) = s.points 0
• Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex 𝕜 P 1) (i : Fin 2) (p : P) : ↑((s.faceOpposite i).orthogonalProjectionSpan p) = s.points i.rev
• Affine.Simplex.orthogonalProjectionSpan_congr 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {m n : ℕ} {s₁ : Affine.Simplex 𝕜 P m} {s₂ : Affine.Simplex 𝕜 P n} {p₁ p₂ : P} (h : Set.range s₁.points = Set.range s₂.points) (hp : p₁ = p₂) : ↑(s₁.orthogonalProjectionSpan p₁) = ↑(s₂.orthogonalProjectionSpan p₂)
• Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} {r₁ : 𝕜} (s : Affine.Simplex 𝕜 P n) {p p₁o : P} (hp₁o : p₁o ∈ affineSpan 𝕜 (Set.range s.points)) : ↑(s.orthogonalProjectionSpan (r₁ • (p -ᵥ ↑(s.orthogonalProjectionSpan p)) +ᵥ p₁o)) = p₁o
• Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) {p₁ : P} (p₂ : P) (r : 𝕜) (hp : p₁ ∈ affineSpan 𝕜 (Set.range s.points)) : s.orthogonalProjectionSpan (r • (p₂ -ᵥ ↑(s.orthogonalProjectionSpan p₂)) +ᵥ p₁) = ⟨p₁, hp⟩
• Affine.Simplex.orthogonalProjectionSpan_reindex 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {m n : ℕ} (s : Affine.Simplex 𝕜 P m) (e : Fin (m + 1) ≃ Fin (n + 1)) (p : P) : ↑((s.reindex e).orthogonalProjectionSpan p) = ↑(s.orthogonalProjectionSpan p)
• Affine.Simplex.orthogonalProjectionSpan_map 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] {V₂ : Type u_4} {P₂ : Type u_5} [NormedAddCommGroup V₂] [InnerProductSpace 𝕜 V₂] [MetricSpace P] [NormedAddTorsor V P] [MetricSpace P₂] [NormedAddTorsor V₂ P₂] {n : ℕ} (s : Affine.Simplex 𝕜 P n) (f : P →ᵃⁱ[𝕜] P₂) (p : P) : ↑((s.map f.toAffineMap ⋯).orthogonalProjectionSpan (f p)) = f ↑(s.orthogonalProjectionSpan p)
• Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) {p₁ : P} (p₂ : P) (hp₁ : p₁ ∈ affineSpan 𝕜 (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)
• Affine.Simplex.orthogonalProjectionSpan_restrict 📋 Mathlib.Geometry.Euclidean.Projection
  {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [RCLike 𝕜] [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex 𝕜 P n) (S : AffineSubspace 𝕜 P) (hS : affineSpan 𝕜 (Set.range s.points) ≤ S) (p : ↥S) : ↑↑((s.restrict S hS).orthogonalProjectionSpan p) = ↑(s.orthogonalProjectionSpan ↑p)
• Affine.Simplex.closedInterior_inter_shift_zero 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [PartialOrder k] [IsOrderedAddMonoid k] [AddCommGroup V] [AddTorsor V P] [Module k V] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) [ZeroLEOneClass k] : s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0) = {s.points i}
• Affine.Simplex.disjoint_closedInterior_shift 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [PartialOrder k] [IsOrderedAddMonoid k] [AddCommGroup V] [AddTorsor V P] [Module k V] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) {x : k} (hx : x < 0 ∨ 1 < x) : Disjoint s.closedInterior ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) x)
• Affine.Simplex.closedInterior_inter_shift_eq_homothety 📋 Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [LinearOrder k] [IsOrderedRing k] [AddCommGroup V] [Module k V] [AddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex k P n) (i : Fin (n + 1)) {x : k} (hx : x ∈ Set.Icc 0 1) : s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) x) = ⇑(AffineMap.homothety (s.points i) x) '' (s.faceOpposite i).closedInterior
• Affine.Simplex.ne_altitudeFoot 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.points i ≠ s.altitudeFoot i
• Affine.Simplex.altitudeFoot_eq_point_rev 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex ℝ P 1) (i : Fin 2) : s.altitudeFoot i = s.points i.rev
• Affine.Simplex.affineSpan_pair_altitudeFoot_eq_altitude 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : line[ℝ, s.altitudeFoot i, s.points i] = s.altitude i
• Affine.Simplex.mem_altitude 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.points i ∈ s.altitude i
• Affine.Simplex.inner_vsub_altitudeFoot_vsub_altitudeFoot_eq_zero 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) {i j : Fin (n + 1)} (h : i ≠ j) : have this := ⋯; inner ℝ (s.points j -ᵥ s.altitudeFoot i) (s.points i -ᵥ s.altitudeFoot i) = 0
• Affine.Simplex.altitudeFoot_mem_affineSpan 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.altitudeFoot i ∈ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.neg_mul_lt_inner_vsub_altitudeFoot 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i j : Fin (n + 1)) : -(s.height i * s.height j) < inner ℝ (s.points i -ᵥ s.altitudeFoot i) (s.points j -ᵥ s.altitudeFoot j)
• Affine.Simplex.height_eq_dist 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex ℝ P 1) (i : Fin 2) : s.height i = dist (s.points 0) (s.points 1)
• Affine.Simplex.vectorSpan_isOrtho_altitude_direction 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : vectorSpan ℝ (s.points '' {i}ᶜ) ⟂ (s.altitude i).direction
• Affine.Simplex.neg_one_lt_inner_vsub_altitudeFoot_div 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [n.AtLeastTwo] (s : Affine.Simplex ℝ P n) (i j : Fin (n + 1)) : -1 < inner ℝ (s.points i -ᵥ s.altitudeFoot i) (s.points j -ᵥ s.altitudeFoot j) / (s.height i * s.height j)
• Affine.Simplex.abs_inner_vsub_altitudeFoot_lt_mul 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] {i j : Fin (n + 1)} (hij : i ≠ j) : |inner ℝ (s.points i -ᵥ s.altitudeFoot i) (s.points j -ᵥ s.altitudeFoot j)| < s.height i * s.height j
• Affine.Simplex.abs_inner_vsub_altitudeFoot_div_lt_one 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] {i j : Fin (n + 1)} (hij : i ≠ j) : |inner ℝ (s.points i -ᵥ s.altitudeFoot i) (s.points j -ᵥ s.altitudeFoot j) / (s.height i * s.height j)| < 1
• Affine.Simplex.inner_vsub_vsub_altitudeFoot_eq_height_sq 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) [NeZero n] {i j : Fin (n + 1)} (h : i ≠ j) : inner ℝ (s.points i -ᵥ s.points j) (s.points i -ᵥ s.altitudeFoot i) = s.height i ^ 2
• Affine.Simplex.altitudeFoot_mem_affineSpan_faceOpposite 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.altitudeFoot i ∈ affineSpan ℝ (Set.range (s.faceOpposite i).points)
• Affine.Simplex.altitudeFoot_mem_affineSpan_image_compl 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.altitudeFoot i ∈ affineSpan ℝ (s.points '' {i}ᶜ)
• Affine.Simplex.direction_altitude 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : (s.altitude i).direction = (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)
• Affine.Simplex.altitude_def 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.altitude i = AffineSubspace.mk' (s.points i) (affineSpan ℝ (s.points '' {i}ᶜ)).directionᗮ ⊓ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.affineSpan_pair_eq_altitude_iff 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) (p : P) : line[ℝ, p, s.points i] = s.altitude i ↔ p ≠ s.points i ∧ p ∈ affineSpan ℝ (Set.range s.points) ∧ p -ᵥ s.points i ∈ (affineSpan ℝ (s.points '' {i}ᶜ)).directionᗮ
• Affine.Simplex.closedInterior_inter_affineSubspaceMk'_lineMap_altitudeFoot 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) (x : ℝ) : s.closedInterior ∩ ↑(AffineSubspace.mk' ((AffineMap.lineMap (s.points i) (s.altitudeFoot i)) x) (s.altitude i).directionᗮ) = s.closedInterior ∩ ↑((affineSpan ℝ (s.points '' {i}ᶜ)).shift (s.points i) x)
• Affine.Simplex.affineSubspaceMk'_lineMap_altitudeFoot_eq_shift 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) (x : ℝ) : AffineSubspace.mk' ((AffineMap.lineMap (s.points i) (s.altitudeFoot i)) x) (s.altitude i).directionᗮ ⊓ affineSpan ℝ (Set.range s.points) = (affineSpan ℝ (s.points '' {i}ᶜ)).shift (s.points i) x
• Affine.Simplex.height_restrict 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (S : AffineSubspace ℝ P) (hS : affineSpan ℝ (Set.range s.points) ≤ S) (i : Fin (n + 1)) : (s.restrict S hS).height i = s.height i
• Affine.Simplex.altitudeFoot_restrict 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (S : AffineSubspace ℝ P) (hS : affineSpan ℝ (Set.range s.points) ≤ S) (i : Fin (n + 1)) : ↑((s.restrict S hS).altitudeFoot i) = s.altitudeFoot i
• Affine.Simplex.map_altitude_restrict 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (S : AffineSubspace ℝ P) (hS : affineSpan ℝ (Set.range s.points) ≤ S) (i : Fin (n + 1)) : AffineSubspace.map S.subtype ((s.restrict S hS).altitude i) = s.altitude i
• Affine.Simplex.altitude_restrict_eq_comap_subtype 📋 Mathlib.Geometry.Euclidean.Altitude
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P n) (S : AffineSubspace ℝ P) (hS : affineSpan ℝ (Set.range s.points) ≤ S) (i : Fin (n + 1)) : (s.restrict S hS).altitude i = AffineSubspace.comap S.subtype (s.altitude i)
• Affine.Simplex.signedInfDist_apply_of_ne 📋 Mathlib.Geometry.Euclidean.SignedDist
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) {i j : Fin (n + 1)} (h : j ≠ i) : (s.signedInfDist i) (s.points j) = 0
• Affine.Simplex.signedInfDist_apply_self 📋 Mathlib.Geometry.Euclidean.SignedDist
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : (s.signedInfDist i) (s.points i) = ‖s.points i -ᵥ ↑((s.faceOpposite i).orthogonalProjectionSpan (s.points i))‖
• Affine.Simplex.abs_signedInfDist_eq_dist_of_mem_affineSpan_range 📋 Mathlib.Geometry.Euclidean.SignedDist
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} (i : Fin (n + 1)) {p : P} (h : p ∈ affineSpan ℝ (Set.range s.points)) : |(s.signedInfDist i) p| = dist p ↑((s.faceOpposite i).orthogonalProjectionSpan p)
• Affine.Simplex.signedInfDist_affineCombination 📋 Mathlib.Geometry.Euclidean.SignedDist
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) {w : Fin (n + 1) → ℝ} (h : ∑ i, w i = 1) : (s.signedInfDist i) ((Finset.affineCombination ℝ Finset.univ s.points) w) = w i * ‖s.points i -ᵥ ↑((s.faceOpposite i).orthogonalProjectionSpan (s.points i))‖
• Affine.Simplex.incenter_ne_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.incenter ≠ s.points i
• Affine.Simplex.touchpoint_eq_point_rev 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Simplex ℝ P 1) (signs : Finset (Fin 2)) (i : Fin 2) : s.touchpoint signs i = s.points i.rev
• Affine.Simplex.ExcenterExists.excenter_ne_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) (i : Fin (n + 1)) : s.excenter signs ≠ s.points i
• Affine.Simplex.ExcenterExists.touchpoint_ne_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} [n.AtLeastTwo] {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) (i j : Fin (n + 1)) : s.touchpoint signs i ≠ s.points j
• Affine.Simplex.touchpoint_empty_ne_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i j : Fin (n + 1)) : s.touchpoint ∅ i ≠ s.points j
• Affine.Simplex.incenter_mem_affineSpan_range 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) : s.incenter ∈ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.eq_touchpoint_of_isTangentAt_exsphere 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))} {i : Fin (n + 1)} {p : P} (ht : (s.exsphere signs).IsTangentAt p (affineSpan ℝ (Set.range (s.faceOpposite i).points))) : p = s.touchpoint signs i
• Affine.Triangle.sbtw_touchpoint_empty 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : Sbtw ℝ (t.points i₁) (Affine.Simplex.touchpoint t ∅ i₂) (t.points i₃)
• Affine.Simplex.ExcenterExists.isTangentAt_touchpoint 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) (i : Fin (n + 1)) : (s.exsphere signs).IsTangentAt (s.touchpoint signs i) (affineSpan ℝ (Set.range (s.faceOpposite i).points))
• Affine.Simplex.isTangentAt_insphere_touchpoint 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.insphere.IsTangentAt (s.touchpoint ∅ i) (affineSpan ℝ (Set.range (s.faceOpposite i).points))
• Affine.Triangle.sbtw_touchpoint_singleton 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : Sbtw ℝ (t.points i₁) (Affine.Simplex.touchpoint t {i₂} i₂) (t.points i₃)
• Affine.Triangle.touchpoint_singleton_sbtw 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : Sbtw ℝ (Affine.Simplex.touchpoint t {i₁} i₂) (t.points i₃) (t.points i₁)
• Affine.Simplex.ExcenterExists.isTangentAt_exsphere_iff_eq_touchpoint 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) {i : Fin (n + 1)} {p : P} : (s.exsphere signs).IsTangentAt p (affineSpan ℝ (Set.range (s.faceOpposite i).points)) ↔ p = s.touchpoint signs i
• Affine.Simplex.isTangentAt_insphere_iff_eq_touchpoint 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {i : Fin (n + 1)} {p : P} : s.insphere.IsTangentAt p (affineSpan ℝ (Set.range (s.faceOpposite i).points)) ↔ p = s.touchpoint ∅ i
• Affine.Simplex.ExcenterExists.excenter_mem_affineSpan_range 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) : s.excenter signs ∈ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.touchpoint_mem_affineSpan_simplex 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (signs : Finset (Fin (n + 1))) (i : Fin (n + 1)) : s.touchpoint signs i ∈ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.sSameSide_incenter_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameSide s.incenter (s.points i)
• Affine.Simplex.sSameSide_point_incenter 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SSameSide (s.points i) s.incenter
• Affine.Triangle.sSameSide_affineSpan_pair_incenter_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : line[ℝ, t.points i₂, t.points i₃].SSameSide (Affine.Simplex.incenter t) (t.points i₁)
• Affine.Triangle.sSameSide_affineSpan_pair_point_incenter 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : line[ℝ, t.points i₂, t.points i₃].SSameSide (t.points i₁) (Affine.Simplex.incenter t)
• Affine.Simplex.excenter_singleton_mem_affineSpan_range 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i : Fin (n + 1)) : s.excenter {i} ∈ affineSpan ℝ (Set.range s.points)
• Affine.Simplex.incenter_notMem_affineSpan_pair 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i j : Fin (n + 1)) : s.incenter ∉ line[ℝ, s.points i, s.points j]
• Affine.Simplex.sum_inv_height_sq_smul_vsub_eq_zero 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) : ∑ i, (s.height i)⁻¹ ^ 2 • (s.points i -ᵥ s.altitudeFoot i) = 0
• Affine.Simplex.incenter_notMem_affineSpan_faceOpposite 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : s.incenter ∉ affineSpan ℝ (Set.range (s.faceOpposite i).points)
• Affine.Simplex.ExcenterExists.affineSpan_faceOpposite_eq_orthRadius 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] {s : Affine.Simplex ℝ P n} [hf : Fact (Module.finrank ℝ V = n)] {signs : Finset (Fin (n + 1))} (h : s.ExcenterExists signs) (i : Fin (n + 1)) : affineSpan ℝ (Set.range (s.faceOpposite i).points) = (s.exsphere signs).orthRadius (s.touchpoint signs i)
• Affine.Simplex.affineSpan_faceOpposite_eq_orthRadius_insphere 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [Fact (Module.finrank ℝ V = n)] (i : Fin (n + 1)) : affineSpan ℝ (Set.range (s.faceOpposite i).points) = s.insphere.orthRadius (s.touchpoint ∅ i)
• Affine.Triangle.affineSpan_pair_eq_orthRadius 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (t : Affine.Triangle ℝ P) [Fact (Module.finrank ℝ V = 2)] (signs : Finset (Fin 3)) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : line[ℝ, t.points i₂, t.points i₃] = (Affine.Simplex.exsphere t signs).orthRadius (Affine.Simplex.touchpoint t signs i₁)
• Affine.Simplex.incenter_notMem_affineSpan_face 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) {fs : Finset (Fin (n + 1))} {m : ℕ} (hfs : fs.card = m + 1) (hne : m ≠ n) : s.incenter ∉ affineSpan ℝ (Set.range (s.face hfs).points)
• Affine.Simplex.sOppSide_excenter_singleton_point 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i : Fin (n + 1)) : (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SOppSide (s.excenter {i}) (s.points i)
• Affine.Simplex.sOppSide_point_excenter_singleton 📋 Mathlib.Geometry.Euclidean.Incenter
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [NeZero n] (s : Affine.Simplex ℝ P n) [n.AtLeastTwo] (i : Fin (n + 1)) : (affineSpan ℝ (Set.range (s.faceOpposite i).points)).SOppSide (s.points i) (s.excenter {i})

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59