EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_of_isDiameter
∀ {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} {s : EuclideanGeometry.Sphere P},
s.IsDiameter p₁ p₃ → (EuclideanGeometry.angle p₁ p₂ p₃ = Real.pi / 2 ↔ p₂ ∈ s)Thales' theorem: The angle inscribed in a semicircle is a right angle.
- Defined in
- Mathlib.Geometry.Euclidean.Angle.Sphere
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
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
- mul_oneproof · cited by 3,885
- InnerProductSpacestatement and proof · cited by 3,523
- add_zeroproof · cited by 2,707
- Real.pistatement and proof · cited by 1,774
- MetricSpacestatement and proof · cited by 1,684
- Dist.distproof · cited by 1,539
- NormedAddTorsorstatement and proof · cited by 1,325
- Inner.innerproof · cited by 1,089
- pow_oneproof · cited by 894
- VSub.vsubproof · cited by 817
Cited by2
Results whose statement or proof uses this declaration.
- EuclideanGeometry.Sphere.thales_theoremproof · cited by 1
- EuclideanGeometry.Sphere.angle_eq_pi_div_two_iff_mem_sphere_ofDiameterproof · cited by 1