Theorems · Theorem · geometry

EuclideanGeometry.Sphere.inv_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 p₃ : P},
  p₁ ∈ s →
    p₂ ∈ s →
      p₃ ∈ s →
        p₁ ≠ p₂ →
          p₁ ≠ p₃ →
            p₂ ≠ p₃ →
              ((EuclideanGeometry.oangle p₁ p₂ p₃).tan⁻¹ / 2) •
                    (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (p₃ -ᵥ p₁) +ᵥ
                  midpoint ℝ p₁ p₃ =
                s.center

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

Defined in

Mathlib.Geometry.Euclidean.Angle.Sphere

Cited by

1 results in Mathlib

Foundations

Depth 291 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented