Theorems · Theorem · geometry

EuclideanGeometry.angle_lt_pi_div_three_of_le_of_le_of_ne

∀ {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] {p₁ p₂ p₃ : P},
  EuclideanGeometry.angle p₁ p₂ p₃ ≤ EuclideanGeometry.angle p₂ p₃ p₁ →
    EuclideanGeometry.angle p₁ p₂ p₃ ≤ EuclideanGeometry.angle p₃ p₁ p₂ →
      EuclideanGeometry.angle p₁ p₂ p₃ ≠ EuclideanGeometry.angle p₂ p₃ p₁ ∨
          EuclideanGeometry.angle p₁ p₂ p₃ ≠ EuclideanGeometry.angle p₃ p₁ p₂ ∨
            EuclideanGeometry.angle p₂ p₃ p₁ ≠ EuclideanGeometry.angle p₃ p₁ p₂ →
        EuclideanGeometry.angle p₁ p₂ p₃ < Real.pi / 3

The least angle of a possibly degenerate triangle is less than π / 3, unless all angles are equal.

Defined in

Mathlib.Geometry.Euclidean.Triangle

Cited by

0 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor