# Midpoint of a segment #

## Main definitions #

• midpoint R x y: midpoint of the segment [x, y]. We define it for x and y in a module over a ring R with invertible 2.
• AddMonoidHom.ofMapMidpoint: construct an AddMonoidHom given a map f such that f sends zero to zero and midpoints to midpoints.

## Main theorems #

• midpoint_eq_iff: z is the midpoint of [x, y] if and only if x + y = z + z,
• midpoint_unique: midpoint R x y does not depend on R;
• midpoint x y is linear both in x and y;
• pointReflection_midpoint_left, pointReflection_midpoint_right: Equiv.pointReflection (midpoint R x y) swaps x and y.

We do not mark most lemmas as @[simp] because it is hard to tell which side is simpler.

## Tags #

def midpoint (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
P

midpoint x y is the midpoint of the segment [x, y].

Equations
Instances For
@[simp]
theorem AffineMap.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [] [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] (f : P →ᵃ[R] P') (a : P) (b : P) :
f (midpoint R a b) = midpoint R (f a) (f b)
@[simp]
theorem AffineEquiv.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [Ring R] [] [] [Module R V] [] [] [Module R V'] [AddTorsor V' P'] (f : P ≃ᵃ[R] P') (a : P) (b : P) :
f (midpoint R a b) = midpoint R (f a) (f b)
theorem AffineEquiv.pointReflection_midpoint_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
@[simp]
theorem Equiv.pointReflection_midpoint_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
theorem midpoint_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
midpoint R x y = midpoint R y x
theorem AffineEquiv.pointReflection_midpoint_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
@[simp]
theorem Equiv.pointReflection_midpoint_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
theorem midpoint_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) (p₃ : P) (p₄ : P) :
midpoint R p₁ p₂ -ᵥ midpoint R p₃ p₄ = midpoint R (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)
theorem midpoint_vadd_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (v : V) (v' : V) (p : P) (p' : P) :
midpoint R v v' +ᵥ midpoint R p p' = midpoint R (v +ᵥ p) (v' +ᵥ p')
theorem midpoint_eq_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} {z : P} :
midpoint R x y = z x = y
@[simp]
theorem midpoint_pointReflection_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
midpoint R () y = x
@[simp]
theorem midpoint_pointReflection_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) (y : P) :
midpoint R y () = x
@[simp]
theorem midpoint_vsub_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) :
midpoint R p₁ p₂ -ᵥ p₁ = 2 (p₂ -ᵥ p₁)
@[simp]
theorem midpoint_vsub_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) :
midpoint R p₁ p₂ -ᵥ p₂ = 2 (p₁ -ᵥ p₂)
@[simp]
theorem left_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) :
p₁ -ᵥ midpoint R p₁ p₂ = 2 (p₁ -ᵥ p₂)
@[simp]
theorem right_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) :
p₂ -ᵥ midpoint R p₁ p₂ = 2 (p₂ -ᵥ p₁)
theorem midpoint_vsub {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) (p : P) :
midpoint R p₁ p₂ -ᵥ p = 2 (p₁ -ᵥ p) + 2 (p₂ -ᵥ p)
theorem vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (p₁ : P) (p₂ : P) (p : P) :
p -ᵥ midpoint R p₁ p₂ = 2 (p -ᵥ p₁) + 2 (p -ᵥ p₂)
@[simp]
theorem midpoint_sub_left {R : Type u_1} {V : Type u_2} [Ring R] [] [] [Module R V] (v₁ : V) (v₂ : V) :
midpoint R v₁ v₂ - v₁ = 2 (v₂ - v₁)
@[simp]
theorem midpoint_sub_right {R : Type u_1} {V : Type u_2} [Ring R] [] [] [Module R V] (v₁ : V) (v₂ : V) :
midpoint R v₁ v₂ - v₂ = 2 (v₁ - v₂)
@[simp]
theorem left_sub_midpoint {R : Type u_1} {V : Type u_2} [Ring R] [] [] [Module R V] (v₁ : V) (v₂ : V) :
v₁ - midpoint R v₁ v₂ = 2 (v₁ - v₂)
@[simp]
theorem right_sub_midpoint {R : Type u_1} {V : Type u_2} [Ring R] [] [] [Module R V] (v₁ : V) (v₂ : V) :
v₂ - midpoint R v₁ v₂ = 2 (v₂ - v₁)
@[simp]
theorem midpoint_eq_left_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} :
midpoint R x y = x x = y
@[simp]
theorem left_eq_midpoint_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} :
x = midpoint R x y x = y
@[simp]
theorem midpoint_eq_right_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} :
midpoint R x y = y x = y
@[simp]
theorem right_eq_midpoint_iff (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} :
y = midpoint R x y x = y
theorem midpoint_eq_midpoint_iff_vsub_eq_vsub (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {x' : P} {y : P} {y' : P} :
midpoint R x y = midpoint R x' y' x -ᵥ x' = y' -ᵥ y
theorem midpoint_eq_iff' (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] {x : P} {y : P} {z : P} :
midpoint R x y = z = y
theorem midpoint_unique (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (R' : Type u_6) [Ring R'] [] [Module R' V] (x : P) (y : P) :
midpoint R x y = midpoint R' x y

midpoint does not depend on the ring R.

@[simp]
theorem midpoint_self (R : Type u_1) {V : Type u_2} {P : Type u_4} [Ring R] [] [] [Module R V] [] (x : P) :
midpoint R x x = x
@[simp]
theorem midpoint_add_self (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) (y : V) :
midpoint R x y + midpoint R x y = x + y
theorem midpoint_zero_add (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) (y : V) :
midpoint R 0 (x + y) = midpoint R x y
theorem midpoint_eq_smul_add (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) (y : V) :
midpoint R x y = 2 (x + y)
@[simp]
theorem midpoint_self_neg (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) :
midpoint R x (-x) = 0
@[simp]
theorem midpoint_neg_self (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) :
midpoint R (-x) x = 0
@[simp]
theorem midpoint_sub_add (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) (y : V) :
midpoint R (x - y) (x + y) = x
@[simp]
theorem midpoint_add_sub (R : Type u_1) {V : Type u_2} [Ring R] [] [] [Module R V] (x : V) (y : V) :
midpoint R (x + y) (x - y) = x
def AddMonoidHom.ofMapMidpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [Ring R] [] [] [Module R E] [Ring R'] [] [] [Module R' F] (f : EF) (h0 : f 0 = 0) (hm : ∀ (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) :
E →+ F

A map f : E → F sending zero to zero and midpoints to midpoints is an AddMonoidHom.

Equations
Instances For
@[simp]
theorem AddMonoidHom.coe_ofMapMidpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [Ring R] [] [] [Module R E] [Ring R'] [] [] [Module R' F] (f : EF) (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