Theorems · Theorem · geometry

Affine.Triangle.circumsphere_eq_of_dist_of_oangle

∀ {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] [hd2 : Fact (Module.finrank ℝ V = 2)] [inst_4 : Module.Oriented ℝ V (Fin 2)]
  (t : Affine.Triangle ℝ P) {i₁ i₂ i₃ : Fin 3},
  i₁ ≠ i₂ →
    i₁ ≠ i₃ →
      i₂ ≠ i₃ →
        Affine.Simplex.circumsphere t =
          {
            center :=
              ((EuclideanGeometry.oangle (t.points i₁) (t.points i₂) (t.points i₃)).tan⁻¹ / 2) •
                  (EuclideanGeometry.o.rotation ↑(Real.pi / 2)) (t.points i₃ -ᵥ t.points i₁) +ᵥ
                midpoint ℝ (t.points i₁) (t.points i₃),
            radius :=
              dist (t.points i₁) (t.points i₃) /
                  |(EuclideanGeometry.oangle (t.points i₁) (t.points i₂) (t.points i₃)).sin| /
                2 }

The circumsphere of a triangle may be expressed explicitly in terms of two points and the angle at the third point.

Defined in

Mathlib.Geometry.Euclidean.Angle.Sphere

Cited by

1 results in Mathlib

Foundations

Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorFactModule.Oriented