This research shows unit area isosceles trapezoids exist in measurable planar sets, indicating broader implications for geometric configurations.
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.
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Junnosuke Koizumi (2025) studied this question.
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