Theorems · Theorem · geometry

EuclideanGeometry.dist_orthogonalProjection_eq_iff_oangle_eq

∀ {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] [inst_4 : Fact (Module.finrank ℝ V = 2)] [inst_5 : Module.Oriented ℝ V (Fin 2)]
  {p p' : P} {s₁ s₂ : AffineSubspace ℝ P} (hp'₁ : p' ∈ s₁) (hp'₂ : p' ∈ s₂),
  ↑((EuclideanGeometry.orthogonalProjection s₁) p) ≠ ↑((EuclideanGeometry.orthogonalProjection s₂) p) →
    ↑((EuclideanGeometry.orthogonalProjection s₁) p) ≠ p' →
      ↑((EuclideanGeometry.orthogonalProjection s₂) p) ≠ p' →
        (dist p ↑((EuclideanGeometry.orthogonalProjection s₁) p) =
            dist p ↑((EuclideanGeometry.orthogonalProjection s₂) p) ↔
          EuclideanGeometry.oangle (↑((EuclideanGeometry.orthogonalProjection s₁) p)) p' p =
            EuclideanGeometry.oangle p p' ↑((EuclideanGeometry.orthogonalProjection s₂) p))

A point p is equidistant to two affine subspaces (typically lines, for this version of the lemma) if and only if the oriented angles at a point p' in their intersection between p and its orthogonal projections onto the subspaces are equal.

Defined in

Mathlib.Geometry.Euclidean.Angle.Bisector

Cited by

0 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented