Theorems · Theorem · geometry

Affine.Triangle.altitude_replace_orthocenter_eq_affineSpan

∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] {t₁ t₂ : Affine.Triangle ℝ P} {i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3},
  i₁ ≠ i₂ →
    i₁ ≠ i₃ →
      i₂ ≠ i₃ →
        j₁ ≠ j₂ →
          j₁ ≠ j₃ →
            j₂ ≠ j₃ →
              t₂.points j₁ = t₁.orthocenter →
                t₂.points j₂ = t₁.points i₂ →
                  t₂.points j₃ = t₁.points i₃ → Affine.Simplex.altitude t₂ j₂ = line[ℝ, t₁.points i₁, t₁.points i₂]

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.

Defined in

Mathlib.Geometry.Euclidean.MongePoint

Cited by

1 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor