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?

Every measurable $A\subseteq \mathbb{R}^2$ with infinite measure contains the vertices of a right-angled triangle of area $1$.

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 $1$.

Note the area condition forces a, b, c to be affinely independent, so no separate non-degeneracy hypothesis is needed.