Theorems · Theorem · geometry

Affine.Simplex.abs_inner_vsub_altitudeFoot_lt_mul

∀ {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] {n : ℕ} (s : Affine.Simplex ℝ P n) [inst_4 : n.AtLeastTwo] {i j : Fin (n + 1)},
  i ≠ j → |inner ℝ (s.points i -ᵥ s.altitudeFoot i) (s.points j -ᵥ s.altitudeFoot j)| < s.height i * s.height j

The inner product of two distinct altitudes has absolute value strictly less than the product of their lengths. Equivalently, neither vector is a multiple of the other; the angle between them is not 0 or π.

Defined in

Mathlib.Geometry.Euclidean.Altitude

Cited by

2 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNat.AtLeastTwo