EuclideanGeometry.inner_vsub_vsub_of_mem_sphere_of_mem_sphere
∀ {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 : P} {s₁ s₂ : EuclideanGeometry.Sphere P},
p₁ ∈ s₁ → p₂ ∈ s₁ → p₁ ∈ s₂ → p₂ ∈ s₂ → inner ℝ (s₂.center -ᵥ s₁.center) (p₂ -ᵥ p₁) = 0Suppose that p₁ and p₂ lie in spheres s₁ and s₂. Then the vector between the centers
of those spheres is orthogonal to that between p₁ and p₂; this is a version of
inner_vsub_vsub_of_dist_eq_of_dist_eq for bundled spheres. (In two dimensions, this says that
the diagonals of a kite are orthogonal.)
- Defined in
- Mathlib.Geometry.Euclidean.Sphere.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- InnerProductSpacestatement and proof · cited by 3,523
- MetricSpacestatement and proof · cited by 1,684
- NormedAddTorsorstatement and proof · cited by 1,325
- Inner.innerstatement · cited by 1,089
- VSub.vsubstatement · cited by 817
- EuclideanGeometry.Spherestatement and proof · cited by 233
- EuclideanGeometry.Sphere.centerstatement · cited by 181
- EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphereproof · cited by 3
- EuclideanGeometry.inner_vsub_vsub_of_dist_eq_of_dist_eqproof · cited by 3
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