/-- Suppose we are given a triangle `t₁`, and replace one of its
vertices by its orthocenter, yielding triangle `t₂` (with vertices not
necessarily listed in the same order).  Then an altitude of `t₂` from
a vertex that was not replaced is the corresponding side of `t₁`. -/
theorem altitude_replace_orthocenter_eq_affineSpan {t₁ t₂ : Triangle ℝ P}
    {i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3} (hi₁₂ : i₁ ≠ i₂) (hi₁₃ : i₁ ≠ i₃) (hi₂₃ : i₂ ≠ i₃) (hj₁₂ : j₁ ≠ j₂)
    (hj₁₃ : j₁ ≠ j₃) (hj₂₃ : j₂ ≠ j₃) (h₁ : t₂.points j₁ = t₁.orthocenter)
    (h₂ : t₂.points j₂ = t₁.points i₂) (h₃ : t₂.points j₃ = t₁.points i₃) :
    t₂.altitude j₂ = line[ℝ, t₁.points i₁, t₁.points i₂] := by
  symm
  rw [← h₂, t₂.affineSpan_pair_eq_altitude_iff]
  rw [h₂]
  use t₁.independent.injective.ne hi₁₂
  have he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points) := by
    refine' ext_of_direction_eq _
      ⟨t₁.points i₃, mem_affineSpan ℝ ⟨j₃, h₃⟩, mem_affineSpan ℝ (Set.mem_range_self _)⟩
    refine' eq_of_le_of_finrank_eq (direction_le (spanPoints_subset_coe_of_subset_coe _)) _
    · have hu : (Finset.univ : Finset (Fin 3)) = {j₁, j₂, j₃} := by
        clear h₁ h₂ h₃
        -- porting note: was `decide!`
        fin_cases j₁ <;> fin_cases j₂ <;> fin_cases j₃
          <;> simp (config := {decide := true}) at hj₁₂ hj₁₃ hj₂₃ ⊢
      rw [← Set.image_univ, ← Finset.coe_univ, hu, Finset.coe_insert, Finset.coe_insert,
        Finset.coe_singleton, Set.image_insert_eq, Set.image_insert_eq, Set.image_singleton, h₁, h₂,
        h₃, Set.insert_subset_iff, Set.insert_subset_iff, Set.singleton_subset_iff]
      exact
        ⟨t₁.orthocenter_mem_affineSpan, mem_affineSpan ℝ (Set.mem_range_self _),
          mem_affineSpan ℝ (Set.mem_range_self _)⟩
    · rw [direction_affineSpan, direction_affineSpan,
        t₁.independent.finrank_vectorSpan (Fintype.card_fin _),
        t₂.independent.finrank_vectorSpan (Fintype.card_fin _)]
  rw [he]
  use mem_affineSpan ℝ (Set.mem_range_self _)
  have hu : Finset.univ.erase j₂ = {j₁, j₃} := by
    clear h₁ h₂ h₃
    -- porting note: was `decide!`
    fin_cases j₁ <;> fin_cases j₂ <;> fin_cases j₃
      <;> simp (config := {decide := true}) at hj₁₂ hj₁₃ hj₂₃ ⊢
  rw [hu, Finset.coe_insert, Finset.coe_singleton, Set.image_insert_eq, Set.image_singleton, h₁, h₃]
  have hle : (t₁.altitude i₃).directionᗮ ≤ line[ℝ, t₁.orthocenter, t₁.points i₃].directionᗮ :=
    Submodule.orthogonal_le (direction_le (affineSpan_orthocenter_point_le_altitude _ _))
  refine' hle ((t₁.vectorSpan_isOrtho_altitude_direction i₃) _)
  have hui : Finset.univ.erase i₃ = {i₁, i₂} := by
    clear hle h₂ h₃
    -- porting note: was `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at hi₁₂ hi₁₃ hi₂₃ ⊢
  

Proof state:
case right
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : points t₂ j₁ = orthocenter t₁
h₂ : points t₂ j₂ = points t₁ i₂
h₃ : points t₂ j₃ = points t₁ i₃
he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points)
hu : erase univ j₂ = {j₁, j₃}
hle : (direction (altitude t₁ i₃))ᗮ ≤ (direction (affineSpan ℝ {orthocenter t₁, points t₁ i₃}))ᗮ
hui : erase univ i₃ = {i₁, i₂}
⊢ points t₁ i₁ -ᵥ points t₁ i₂ ∈ vectorSpan ℝ (t₁.points '' ↑(erase univ i₃))

Next tactic:
rw [hui, Finset.coe_insert, Finset.coe_singleton, Set.image_insert_eq, Set.image_singleton]