Finite-nine common-radius circle placement

/-
Copyright (c) 2026 Adam McKenna. All rights reserved.
Released under GPL-3.0-or-later as described in the file LICENSE.
Authors: Adam McKenna
-/

import Definitions.Def_Erdos9796FiniteNine_N8Interface

/-! Statement-only transfer node for finite-nine common-radius circle placement.

SKETCH — NOT PROMOTABLE. The source project already proves this result; the proof
solution will be transferred separately after this public theorem node exists.
-/

open scoped EuclideanGeometry

theorem Erdos9796FiniteNine.circle_placement {A : Finset ℝ²}
    (S : Batch3N9.Problem97.FiniteEndpointShell A)
    (hN4e : S.N4eCapContainment) :
    ∃ d : ℝ, 0 < d ∧
      (∀ x ∈ S.CP.C1, dist S.triangle.v1 x = d) ∧
      (∀ x ∈ S.CP.C2, dist S.triangle.v2 x = d) ∧
      (∀ x ∈ S.CP.C3, dist S.triangle.v3 x = d) := by sorry