putnam-1965-b6

import Mathlib

open EuclideanGeometry Topology Filter Complex SimpleGraph.Walk

theorem putnam_1965_b6 (A B C D : EuclideanSpace ℝ (Fin 2))
    (S : Set (EuclideanSpace ℝ (Fin 2)))
    (hS : S = {A, B, C, D})
    (hdistinct : S.ncard = 4)
    (through : (ℝ × (EuclideanSpace ℝ (Fin 2))) → (EuclideanSpace ℝ (Fin 2)) → Prop)
    (through_def : through = fun (r, P) => fun Q => dist P Q = r)
    (hABCD : ∀ r s : ℝ, ∀ P Q,
      through (r, P) A ∧ through (r, P) B ∧ through (s, Q) C ∧ through (s, Q) D →
      ∃ I, through (r, P) I ∧ through (s, Q) I) :
    Collinear ℝ S ∨ ∃ r : ℝ, ∃ P, ∀ Q ∈ S, through (r, P) Q := by
  sorry