Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure. Must $A$ contain the vertices of an isosceles trapezoid of area $1$? What about an isosceles triangle, or a right-angled triangle, or a cyclic quadrilateral, or a convex polygon with congruent sides?

1True ↔2  ∀ (A : Set (EuclideanSpace ℝ (Fin 2))),3    MeasurableSet A →4      MeasureTheory.volume A = ⊤ →5        ∃ a ∈ A,6          ∃ b ∈ A,7            ∃ c ∈ A,8              ∃ d ∈ A,9                EuclideanGeometry.IsIsoscelesTrapezoid a b c d ∧ MeasureTheory.volume ((convexHull ℝ) {a, b, c, d}) = 1