Theorems · Theorem · geometry

EuclideanGeometry.Sphere.tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center

∀ {V : Type u_3} {P : Type u_4} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] [hd2 : Fact (Module.finrank ℝ V = 2)] [inst_4 : Module.Oriented ℝ V (Fin 2)]
  {s : EuclideanGeometry.Sphere P} {p₁ p₂ : P},
  p₁ ∈ s →
    p₂ ∈ s →
      p₁ ≠ p₂ →
        ((EuclideanGeometry.oangle p₂ p₁ s.center).tan / 2) •
              (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (p₂ -ᵥ p₁) +ᵥ
            midpoint ℝ p₁ p₂ =
          s.center

Given two points on a circle, the center of that circle may be expressed explicitly as a multiple (by half the tangent of the angle between the chord and the radius at one of those points) of a π / 2 rotation of the vector between those points, plus the midpoint of those points.

Defined in

Mathlib.Geometry.Euclidean.Angle.Sphere

Cited by

2 results in Mathlib

Foundations

Depth 290 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented