∀ {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} [inst_4 : NeZero n] {s : Affine.Simplex ℝ P n} {signs : Finset (Fin (n + 1))},
  s.ExcenterExists signs →
    ∀ {i j : Fin (n + 1)},
      i ≠ j →
        ∀ {r : ℝ},
          r ∈ Set.Icc 0 1 →
            SignType.sign ((s.signedInfDist j) ((AffineMap.lineMap (s.excenter signs) (s.touchpoint signs i)) r)) =
              SignType.sign ((s.signedInfDist j) (s.excenter signs))