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?

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?

Koizumi [Ko25] has resolved this question, proving that any set with infinite measure must contain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled triangle, all of area $1$.

This statement formalizes the leading question, for isosceles trapezoids; the remaining configurations are given as variants below. The area of a polygon is taken to be the Lebesgue measure of the convex hull of its vertices.