/-- In a tetrahedron with vertices `x`, `y`, `p`, `q`, any segment `[u, v]` joining the opposite
edges `[x, p]` and `[y, q]` passes through any triangle of vertices `p`, `q`, `z` where
`z ∈ [x, y]`. -/
theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
    (hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
    ¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by
  rw [not_disjoint_iff]
  obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
  obtain rfl | haz' := haz.eq_or_lt
  · rw [zero_add] at habz
    rw [zero_smul, zero_add, habz, one_smul]
    refine' ⟨v, by apply right_mem_segment, segment_subset_convexHull _ _ hv⟩ <;> simp
  obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
  obtain rfl | hav' := hav.eq_or_lt
  · rw [zero_add] at habv
    rw [zero_smul, zero_add, habv, one_smul]
    exact ⟨q, right_mem_segment _ _ _, subset_convexHull _ _ <| by simp⟩
  obtain ⟨au, bu, hau, hbu, habu, rfl⟩ := hu
  have hab : 0 < az * av + bz * au :=
    add_pos_of_pos_of_nonneg (mul_pos haz' hav') (mul_nonneg hbz hau)
  refine'
    ⟨(az * av / (az * av + bz * au)) • (au • x + bu • p) +
        (bz * au / (az * av + bz * au)) • (av • y + bv • q),
      ⟨_, _, _, _, _, rfl⟩, _⟩
  · exact div_nonneg (mul_nonneg haz hav) hab.le
  · exact div_nonneg (mul_nonneg hbz hau) hab.le
  · rw [← add_div, div_self hab.ne']
  rw [smul_add, smul_add, add_add_add_comm, add_comm, ← mul_smul, ← mul_smul]
  classical
    let w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
    let z : Fin 3 → E := ![p, q, az • x + bz • y]
    have hw₀ : ∀ i, 0 ≤ w i := by
      rintro i
      fin_cases i
      · exact mul_nonneg (mul_nonneg haz hav) hbu
      · exact mul_nonneg (mul_nonneg hbz hau) hbv
      · exact mul_nonneg hau hav
    have hw : ∑ i, w i = az * av + bz * au := by
      trans az * av * bu + (bz * au * bv + au * av)
      · simp [Fin.sum_univ_succ, Fin.sum_univ_zero]
      rw [← one_mul (au * av), ← habz, add_mul, ← add_assoc, add_add_add_comm, mul_assoc, ← mul_add,
        mul_assoc, ← mul_add, mul_comm av, ← add_mul, ← mul_add, add_comm bu, add_comm bv, habu,
        habv, one_mul, mul_one]
    have hz : ∀ i, z i ∈ ({p, q, az • x + bz • y} : Set E) := fun i => by fin_cases i <;> simp
    convert Finset.centerMass_mem_convexHull (Finset.univ : Finset (Fin 3)) (fun i _ => hw₀ i)
        (by rwa [hw]) fun i _ => hz i
    rw [Finset.centerMass]
    simp_rw [div_eq_inv_mul, hw, mul_assoc, mul_smul (az * av + bz * au)⁻¹, ← smul_add, add_assoc, ←
      mul_assoc]
    

Proof state:
case h.e'_4
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
z : Fin 3 → E := ![p, q, az • x + bz • y]
hw₀ : ∀ (i : Fin 3), 0 ≤ w i
hw : ∑ i : Fin 3, w i = az * av + bz * au
hz : ∀ (i : Fin 3), z i ∈ {p, q, az • x + bz • y}
⊢ (az * av + bz * au)⁻¹ • ((az * av * bu) • p + ((bz * au * bv) • q + ((az * av) • au • x + (bz * au) • av • y))) =
    (az * av + bz * au)⁻¹ •
      ∑ x_1 : Fin 3,
        Matrix.vecCons (az * av * bu) ![bz * au * bv, au * av] x_1 • Matrix.vecCons p ![q, az • x + bz • y] x_1

Next tactic:
congr 3