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Result

Found 109 declarations mentioning midpoint.

• midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : P
• midpoint_self 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x : P) : midpoint R x x = x
• left_eq_midpoint_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} : x = midpoint R x y ↔ x = y
• midpoint_eq_left_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} : midpoint R x y = x ↔ x = y
• midpoint_eq_right_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} : midpoint R x y = y ↔ x = y
• right_eq_midpoint_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y : P} : y = midpoint R x y ↔ x = y
• midpoint_comm 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : midpoint R x y = midpoint R y x
• midpoint_pointReflection_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : midpoint R ((Equiv.pointReflection x) y) y = x
• midpoint_pointReflection_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : midpoint R y ((Equiv.pointReflection x) y) = x
• Equiv.pointReflection_midpoint_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : (Equiv.pointReflection (midpoint R x y)) x = y
• Equiv.pointReflection_midpoint_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : (Equiv.pointReflection (midpoint R x y)) y = x
• midpoint_neg_self 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x : V) : midpoint R (-x) x = 0
• midpoint_self_neg 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x : V) : midpoint R x (-x) = 0
• midpoint_add_sub 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V) : midpoint R (x + y) (x - y) = x
• midpoint_sub_add 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V) : midpoint R (x - y) (x + y) = x
• midpoint_eq_iff' 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} : midpoint R x y = z ↔ (Equiv.pointReflection z) x = y
• midpoint_eq_midpoint_iff_vsub_eq_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x x' y y' : P} : midpoint R x y = midpoint R x' y' ↔ x -ᵥ x' = y' -ᵥ y
• midpoint_zero_add 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V) : midpoint R 0 (x + y) = midpoint R x y
• midpoint_add_self 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V) : midpoint R x y + midpoint R x y = x + y
• AffineEquiv.midpoint_pointReflection_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : midpoint R ((AffineEquiv.pointReflection R x) y) y = x
• AffineEquiv.midpoint_pointReflection_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : midpoint R y ((AffineEquiv.pointReflection R x) y) = x
• AffineEquiv.pointReflection_midpoint_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : (AffineEquiv.pointReflection R (midpoint R x y)) x = y
• AffineEquiv.pointReflection_midpoint_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (x y : P) : (AffineEquiv.pointReflection R (midpoint R x y)) y = x
• midpoint_vsub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p₃ p₄ : P) : midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)
• midpoint_eq_iff 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] {x y z : P} : midpoint R x y = z ↔ (AffineEquiv.pointReflection R z) x = y
• midpoint_unique 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (R' : Type u_6) [Ring R'] [Invertible 2] [Module R' V] (x y : P) : midpoint R x y = midpoint R' x y
• midpoint_eq_smul_add 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (x y : V) : midpoint R x y = ⅟2 • (x + y)
• left_vsub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P) : p₁ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₁ -ᵥ p₂)
• midpoint_vsub_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₁ = ⅟2 • (p₂ -ᵥ p₁)
• midpoint_vsub_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₂ = ⅟2 • (p₁ -ᵥ p₂)
• right_vsub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ : P) : p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ p₁)
• left_sub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V) : v₁ - midpoint R v₁ v₂ = ⅟2 • (v₁ - v₂)
• midpoint_sub_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V) : midpoint R v₁ v₂ - v₁ = ⅟2 • (v₂ - v₁)
• midpoint_sub_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V) : midpoint R v₁ v₂ - v₂ = ⅟2 • (v₁ - v₂)
• right_sub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] (v₁ v₂ : V) : v₂ - midpoint R v₁ v₂ = ⅟2 • (v₂ - v₁)
• AddMonoidHom.ofMapMidpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [Ring R] [Invertible 2] [AddCommGroup E] [Module R E] [Ring R'] [Invertible 2] [AddCommGroup F] [Module R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) : E →+ F
• midpoint_vsub_midpoint_same_left 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p₃ : P) : midpoint R p₁ p₂ -ᵥ midpoint R p₁ p₃ = ⅟2 • (p₂ -ᵥ p₃)
• midpoint_vsub_midpoint_same_right 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p₃ : P) : midpoint R p₁ p₃ -ᵥ midpoint R p₂ p₃ = ⅟2 • (p₁ -ᵥ p₂)
• AffineMap.map_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] (f : P →ᵃ[R] P') (a b : P) : f (midpoint R a b) = midpoint R (f a) (f b)
• midpoint_vadd_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (v v' : V) (p p' : P) : midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p')
• AddMonoidHom.coe_ofMapMidpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [Ring R] [Invertible 2] [AddCommGroup E] [Module R E] [Ring R'] [Invertible 2] [AddCommGroup F] [Module R' F] (f : E → F) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) : ⇑(AddMonoidHom.ofMapMidpoint R R' f h0 hm) = f
• AffineEquiv.map_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] [AddCommGroup V'] [Module R V'] [AddTorsor V' P'] (f : P ≃ᵃ[R] P') (a b : P) : f (midpoint R a b) = midpoint R (f a) (f b)
• midpoint_vsub 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p : P) : midpoint R p₁ p₂ -ᵥ p = ⅟2 • (p₁ -ᵥ p) + ⅟2 • (p₂ -ᵥ p)
• vsub_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Midpoint
  {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (p₁ p₂ p : P) : p -ᵥ midpoint R p₁ p₂ = ⅟2 • (p -ᵥ p₁) + ⅟2 • (p -ᵥ p₂)
• midpoint_mem_openSegment 📋 Mathlib.Analysis.Convex.Segment
  {𝕜 : Type u_1} {E : Type u_2} [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] (x y : E) : midpoint 𝕜 x y ∈ openSegment 𝕜 x y
• midpoint_mem_segment 📋 Mathlib.Analysis.Convex.Segment
  {𝕜 : Type u_1} {E : Type u_2} [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] (x y : E) : midpoint 𝕜 x y ∈ segment 𝕜 x y
• Convex.midpoint_mem 📋 Mathlib.Analysis.Convex.Basic
  {𝕜 : Type u_1} {E : Type u_2} [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [Invertible 2] {s : Set E} {x y : E} (h : Convex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) : midpoint 𝕜 x y ∈ s
• Filter.Tendsto.midpoint 📋 Mathlib.Topology.Algebra.Affine
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [AddCommGroup V] [TopologicalSpace V] [AddTorsor V P] [TopologicalSpace P] [IsTopologicalAddTorsor P] [Ring R] [Module R V] [TopologicalSpace R] [ContinuousSMul R V] {α : Type u_6} {l : Filter α} [Invertible 2] {f₁ f₂ : α → P} {p₁ p₂ : P} (h₁ : Filter.Tendsto f₁ l (nhds p₁)) (h₂ : Filter.Tendsto f₂ l (nhds p₂)) : Filter.Tendsto (fun x => midpoint R (f₁ x) (f₂ x)) l (nhds (midpoint R p₁ p₂))
• AffineIsometryEquiv.pointReflection_midpoint_left 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {V : Type u_2} {P : Type u_10} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x y : P) : (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) x = y
• AffineIsometryEquiv.pointReflection_midpoint_right 📋 Mathlib.Analysis.Normed.Affine.Isometry
  {V : Type u_2} {P : Type u_10} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedSpace ℝ V] (x y : P) : (AffineIsometryEquiv.pointReflection ℝ (midpoint ℝ x y)) y = x
• dist_left_midpoint_eq_dist_right_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : dist p₁ (midpoint 𝕜 p₁ p₂) = dist p₂ (midpoint 𝕜 p₁ p₂)
• dist_midpoint_midpoint_le 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (p₁ p₂ p₃ p₄ : V) : dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2
• dist_left_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : dist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂
• dist_midpoint_left 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : dist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖⁻¹ * dist p₁ p₂
• dist_midpoint_right 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂
• dist_right_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : dist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂
• nndist_midpoint_midpoint_le 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} [SeminormedAddCommGroup V] [NormedSpace ℝ V] (p₁ p₂ p₃ p₄ : V) : nndist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / 2
• nndist_left_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist p₁ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_midpoint_left 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist (midpoint 𝕜 p₁ p₂) p₁ = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_midpoint_right 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖₊⁻¹ * nndist p₁ p₂
• nndist_right_midpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ : P) : nndist p₂ (midpoint 𝕜 p₁ p₂) = ‖2‖₊⁻¹ * nndist p₁ p₂
• AffineMap.ofMapMidpoint 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} {W : Type u_3} {Q : Type u_4} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [NormedAddCommGroup W] [MetricSpace Q] [NormedAddTorsor W Q] [NormedSpace ℝ V] [NormedSpace ℝ W] (f : P → Q) (h : ∀ (x y : P), f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)) (hfc : Continuous f) : P →ᵃ[ℝ] Q
• dist_midpoint_midpoint_le' 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ p₃ p₄ : P) : dist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / ‖2‖
• nndist_midpoint_midpoint_le' 📋 Mathlib.Analysis.Normed.Affine.AddTorsor
  {V : Type u_1} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {𝕜 : Type u_5} [NormedField 𝕜] [NormedSpace 𝕜 V] [Invertible 2] (p₁ p₂ p₃ p₄ : P) : nndist (midpoint 𝕜 p₁ p₂) (midpoint 𝕜 p₃ p₄) ≤ (nndist p₁ p₃ + nndist p₂ p₄) / ‖2‖₊
• midpoint_le_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Ordered
  {k : Type u_1} {E : Type u_2} [Ring k] [LinearOrder k] [IsStrictOrderedRing k] [AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module k E] [IsStrictOrderedModule k E] {a a' b b' : E} [Invertible 2] (ha : a ≤ a') (hb : b ≤ b') : midpoint k a b ≤ midpoint k a' b'
• left_le_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Ordered
  {k : Type u_1} {E : Type u_2} [Field k] [LinearOrder k] [IsStrictOrderedRing k] [AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module k E] [IsStrictOrderedModule k E] [PosSMulReflectLE k E] {a b : E} : a ≤ midpoint k a b ↔ a ≤ b
• midpoint_le_left 📋 Mathlib.LinearAlgebra.AffineSpace.Ordered
  {k : Type u_1} {E : Type u_2} [Field k] [LinearOrder k] [IsStrictOrderedRing k] [AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module k E] [IsStrictOrderedModule k E] [PosSMulReflectLE k E] {a b : E} : midpoint k a b ≤ a ↔ b ≤ a
• midpoint_le_right 📋 Mathlib.LinearAlgebra.AffineSpace.Ordered
  {k : Type u_1} {E : Type u_2} [Field k] [LinearOrder k] [IsStrictOrderedRing k] [AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module k E] [IsStrictOrderedModule k E] [PosSMulReflectLE k E] {a b : E} : midpoint k a b ≤ b ↔ a ≤ b
• right_le_midpoint 📋 Mathlib.LinearAlgebra.AffineSpace.Ordered
  {k : Type u_1} {E : Type u_2} [Field k] [LinearOrder k] [IsStrictOrderedRing k] [AddCommGroup E] [PartialOrder E] [IsOrderedAddMonoid E] [Module k E] [IsStrictOrderedModule k E] [PosSMulReflectLE k E] {a b : E} : b ≤ midpoint k a b ↔ b ≤ a
• Urysohns.CU.lim_eq_midpoint 📋 Mathlib.Topology.UrysohnsLemma
  {X : Type u_1} [TopologicalSpace X] {P : Set X → Set X → Prop} (c : Urysohns.CU P) (x : X) : c.lim x = midpoint ℝ (c.left.lim x) (c.right.lim x)
• wbtw_midpoint 📋 Mathlib.Analysis.Convex.Between
  (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] (x y : P) : Wbtw R x (midpoint R x y) y
• sbtw_midpoint_of_ne 📋 Mathlib.Analysis.Convex.Between
  (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] {x y : P} (h : x ≠ y) : Sbtw R x (midpoint R x y) y
• eq_midpoint_of_dist_eq_half 📋 Mathlib.Analysis.Convex.StrictConvexBetween
  {E : Type u_3} {PE : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] {x y z : PE} (hx : dist x y = dist x z / 2) (hy : dist y z = dist x z / 2) : y = midpoint ℝ x z
• IsometryEquiv.midpoint_fixed 📋 Mathlib.Analysis.Normed.Affine.MazurUlam
  {E : Type u_1} {PE : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] {x y : PE} (e : PE ≃ᵢ PE) : e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y
• IsometryEquiv.map_midpoint 📋 Mathlib.Analysis.Normed.Affine.MazurUlam
  {E : Type u_1} {PE : Type u_2} {F : Type u_3} {PF : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE] [NormedAddTorsor E PE] [NormedAddCommGroup F] [NormedSpace ℝ F] [MetricSpace PF] [NormedAddTorsor F PF] (f : PE ≃ᵢ PF) (x y : PE) : f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
• EuclideanGeometry.angle_midpoint_eq_pi 📋 Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ : P) (hp₁p₂ : p₁ ≠ p₂) : EuclideanGeometry.angle p₁ (midpoint ℝ p₁ p₂) p₂ = Real.pi
• EuclideanGeometry.angle_left_midpoint_eq_pi_div_two_of_dist_eq 📋 Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : dist p₃ p₁ = dist p₃ p₂) : EuclideanGeometry.angle p₃ (midpoint ℝ p₁ p₂) p₁ = Real.pi / 2
• EuclideanGeometry.angle_right_midpoint_eq_pi_div_two_of_dist_eq 📋 Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : dist p₃ p₁ = dist p₃ p₂) : EuclideanGeometry.angle p₃ (midpoint ℝ p₁ p₂) p₂ = Real.pi / 2
• EuclideanGeometry.oangle_midpoint_left 📋 Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p₁ p₂ p₃ : P) : EuclideanGeometry.oangle (midpoint ℝ p₁ p₂) p₂ p₃ = EuclideanGeometry.oangle p₁ p₂ p₃
• EuclideanGeometry.oangle_midpoint_rev_left 📋 Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p₁ p₂ p₃ : P) : EuclideanGeometry.oangle (midpoint ℝ p₂ p₁) p₂ p₃ = EuclideanGeometry.oangle p₁ p₂ p₃
• EuclideanGeometry.oangle_midpoint_rev_right 📋 Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p₁ p₂ p₃ : P) : EuclideanGeometry.oangle p₁ p₂ (midpoint ℝ p₂ p₃) = EuclideanGeometry.oangle p₁ p₂ p₃
• EuclideanGeometry.oangle_midpoint_right 📋 Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (p₁ p₂ p₃ : P) : EuclideanGeometry.oangle p₁ p₂ (midpoint ℝ p₃ p₂) = EuclideanGeometry.oangle p₁ p₂ p₃
• EuclideanGeometry.dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint 📋 Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {p₁ p₂ p : P} (h : p₁ ≠ p₂) : dist p₁ p = dist p₂ p ↔ ∃ r, r • (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = p
• AffineSubspace.midpoint_mem_perpBisector 📋 Mathlib.Geometry.Euclidean.PerpBisector
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ : P) : midpoint ℝ p₁ p₂ ∈ AffineSubspace.perpBisector p₁ p₂
• AffineSubspace.mem_perpBisector_iff_inner_eq_zero 📋 Mathlib.Geometry.Euclidean.PerpBisector
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c p₁ p₂ : P} : c ∈ AffineSubspace.perpBisector p₁ p₂ ↔ inner ℝ (c -ᵥ midpoint ℝ p₁ p₂) (p₂ -ᵥ p₁) = 0
• AffineSubspace.mem_perpBisector_iff_inner_eq_zero' 📋 Mathlib.Geometry.Euclidean.PerpBisector
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {c p₁ p₂ : P} : c ∈ AffineSubspace.perpBisector p₁ p₂ ↔ inner ℝ (p₂ -ᵥ p₁) (c -ᵥ midpoint ℝ p₁ p₂) = 0
• EuclideanGeometry.Sphere.IsDiameter.midpoint_eq_center 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (self : s.IsDiameter p₁ p₂) : midpoint ℝ p₁ p₂ = s.center
• EuclideanGeometry.Sphere.IsDiameter.mk 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (left_mem : p₁ ∈ s) (midpoint_eq_center : midpoint ℝ p₁ p₂ = s.center) : s.IsDiameter p₁ p₂
• EuclideanGeometry.Sphere.isDiameter_iff_left_mem_and_midpoint_eq_center 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} : s.IsDiameter p₁ p₂ ↔ p₁ ∈ s ∧ midpoint ℝ p₁ p₂ = s.center
• EuclideanGeometry.Sphere.isDiameter_iff_right_mem_and_midpoint_eq_center 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [NormedSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} : s.IsDiameter p₁ p₂ ↔ p₂ ∈ s ∧ midpoint ℝ p₁ p₂ = s.center
• EuclideanGeometry.Sphere.dist_center_midpoint_lt_radius 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : dist s.center (midpoint ℝ p₁ p₂) < s.radius
• EuclideanGeometry.Sphere.inner_vsub_center_midpoint_vsub 📋 Mathlib.Geometry.Euclidean.Sphere.Basic
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ : P} {s : EuclideanGeometry.Sphere P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) : inner ℝ (s.center -ᵥ midpoint ℝ p₁ p₂) (p₂ -ᵥ p₁) = 0
• EuclideanGeometry.Sphere.tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center 📋 Mathlib.Geometry.Euclidean.Angle.Sphere
  {V : Type u_3} {P : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) : ((EuclideanGeometry.oangle p₂ p₁ s.center).tan / 2) • (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = s.center
• EuclideanGeometry.Sphere.inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center 📋 Mathlib.Geometry.Euclidean.Angle.Sphere
  {V : Type u_3} {P : Type u_4} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] {s : EuclideanGeometry.Sphere P} {p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂) (hp₁p₃ : p₁ ≠ p₃) (hp₂p₃ : p₂ ≠ p₃) : ((EuclideanGeometry.oangle p₁ p₂ p₃).tan⁻¹ / 2) • (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (p₃ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₃ = s.center
• Affine.Triangle.inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_circumcenter 📋 Mathlib.Geometry.Euclidean.Angle.Sphere
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : ((EuclideanGeometry.oangle (t.points i₁) (t.points i₂) (t.points i₃)).tan⁻¹ / 2) • (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (t.points i₃ -ᵥ t.points i₁) +ᵥ midpoint ℝ (t.points i₁) (t.points i₃) = Affine.Simplex.circumcenter t
• Affine.Triangle.circumsphere_eq_of_dist_of_oangle 📋 Mathlib.Geometry.Euclidean.Angle.Sphere
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)] (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) : Affine.Simplex.circumsphere t = { center := ((EuclideanGeometry.oangle (t.points i₁) (t.points i₂) (t.points i₃)).tan⁻¹ / 2) • (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (t.points i₃ -ᵥ t.points i₁) +ᵥ midpoint ℝ (t.points i₁) (t.points i₃), radius := dist (t.points i₁) (t.points i₃) / |(EuclideanGeometry.oangle (t.points i₁) (t.points i₂) (t.points i₃)).sin| / 2 }
• EuclideanGeometry.dist_sq_add_dist_sq_eq_two_mul_dist_midpoint_sq_add_half_dist_sq 📋 Mathlib.Geometry.Euclidean.Triangle
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (a b c : P) : dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2)
• EuclideanGeometry.dist_sq_add_dist_sq_eq_dist_sq_add_dist_sq_of_angle_eq_pi_div_two 📋 Mathlib.Geometry.Euclidean.BritishFlag
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {a b c d : P} (h : midpoint ℝ a c = midpoint ℝ b d) (ha : EuclideanGeometry.angle b a d = Real.pi / 2) (p : P) : dist p a ^ 2 + dist p c ^ 2 = dist p b ^ 2 + dist p d ^ 2
• EuclideanGeometry.dist_sq_add_dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two 📋 Mathlib.Geometry.Euclidean.BritishFlag
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {a b c d : P} (h : midpoint ℝ a c = midpoint ℝ b d) (p : P) : dist p a ^ 2 + dist p c ^ 2 = dist p b ^ 2 + dist p d ^ 2 ↔ EuclideanGeometry.angle b a d = Real.pi / 2
• EuclideanGeometry.dist_sq_add_dist_sq_eq_dist_sq_add_dist_sq_add_two_mul_dist_mul_dist_mul_cos_angle 📋 Mathlib.Geometry.Euclidean.BritishFlag
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {a b c d : P} (h : midpoint ℝ a c = midpoint ℝ b d) (p : P) : dist p a ^ 2 + dist p c ^ 2 = dist p b ^ 2 + dist p d ^ 2 + 2 * dist b a * dist d a * Real.cos (EuclideanGeometry.angle b a d)
• EuclideanGeometry.dist_sq_add_dist_sq_eq_dist_sq_add_dist_sq_add_two_mul_inner 📋 Mathlib.Geometry.Euclidean.BritishFlag
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {a b c d : P} (h : midpoint ℝ a c = midpoint ℝ b d) (p : P) : dist p a ^ 2 + dist p c ^ 2 = dist p b ^ 2 + dist p d ^ 2 + 2 * inner ℝ (b -ᵥ a) (d -ᵥ a)
• Affine.Triangle.eulerPoint_eq_midpoint 📋 Mathlib.Geometry.Euclidean.NinePointCircle
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (s : Affine.Triangle ℝ P) (i : Fin 3) : Affine.Simplex.eulerPoint s i = midpoint ℝ s.orthocenter (s.points i)
• Affine.Simplex.midpoint_faceOppositeCentroid_eulerPoint 📋 Mathlib.Geometry.Euclidean.NinePointCircle
  {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {n : ℕ} [hn : NeZero n] (s : Affine.Simplex ℝ P n) (i : Fin (n + 1)) : midpoint ℝ (s.faceOppositeCentroid i) (s.eulerPoint i) = s.ninePointCircle.center
• pi_midpoint_apply 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {k : Type u_1} {ι : Type u_2} {V : ι → Type u_3} {P : ι → Type u_4} [Ring k] [Invertible 2] [(i : ι) → AddCommGroup (V i)] [(i : ι) → Module k (V i)] [(i : ι) → AddTorsor (V i) (P i)] (f g : (i : ι) → P i) (i : ι) : midpoint k f g i = midpoint k (f i) (g i)
• homothety_inv_two 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [CharZero k] [AddCommGroup V] [Module k V] [AddTorsor V P] (a b : P) : (AffineMap.homothety a 2⁻¹) b = midpoint k a b
• homothety_one_half 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {k : Type u_1} {V : Type u_2} {P : Type u_3} [Field k] [CharZero k] [AddCommGroup V] [Module k V] [AddTorsor V P] (a b : P) : (AffineMap.homothety a (1 / 2)) b = midpoint k a b
• lineMap_inv_two 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing R] [CharZero R] [AddCommGroup V] [Module R V] [AddTorsor V P] (a b : P) : (AffineMap.lineMap a b) 2⁻¹ = midpoint R a b
• homothety_invOf_two 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [CommRing R] [Invertible 2] [AddCommGroup V] [Module R V] [AddTorsor V P] (a b : P) : (AffineMap.homothety a ⅟2) b = midpoint R a b
• lineMap_one_half 📋 Mathlib.LinearAlgebra.AffineSpace.MidpointZero
  {R : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing R] [CharZero R] [AddCommGroup V] [Module R V] [AddTorsor V P] (a b : P) : (AffineMap.lineMap a b) (1 / 2) = midpoint R a b

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 ce5dd8c