Isosceles trapezoids of unit area with vertices in sets of infinite planar measure

Paul Erdős posed the question of whether every measurable planar set of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of unit area. In this paper, we provide an affirmative answer to this question. Additionally, we present affirmative solutions to similar questions by Erdős concerning isosceles triangles and right-angled triangles.