Theorems · Definition · geometry

Affine.Simplex.ninePointCircle

{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] → {n : ℕ} → Affine.Simplex ℝ P n → EuclideanGeometry.Sphere P

The 3(n+1)-point sphere of a simplex. Due to the lack of a better name and to avoid numbers in the identifier, we still use the name "nine-point circle" even for higher dimensions. The center $N$ is defined on the Euler line, collinear with circumcenter $O$ and centroid $G$, in the order of $O$, $G$, and $N$, with $OG : GN = n : 1$. The radius is $1/n$ of the circumradius.

Defined in

Mathlib.Geometry.Euclidean.NinePointCircle

Cited by

13 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor