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Claim. Let $S\subseteq\mathbb{R}^2$ be measurable. If $S$ is unbounded and has positive Lebesgue measure, it contains the three vertices of an isosceles triangle of area $1$ and the three vertices of a right triangle of area $1$ (Theorem 1). If $S$ has infinite Lebesgue measure, it contains the four vertices of an isosceles trapezoid of area $1$ (Theorem 2). The trapezoid statement does not extend to unbounded sets of positive measure: a small disk about the origin together with the points $(n,0)$ for $n=1,2,3,\dots$ is a counterexample, as the paper remarks.

Covers. Three of the five questions in the problem's statement, all answered yes: the isosceles trapezoid of area $1$, which Theorem 2 gives, and the isosceles triangle and the right triangle, which Theorem 1 gives. The cyclic quadrilateral (yes) and the convex polygon with congruent sides (no) are Kovač and Predojević's, recorded on their claim page; the two pages together settle every part of the statement, one of them negatively.

Method. The proof builds an area-preserving map $f$ of the plane that rotates each circle about the origin by an angle $\varphi(r)=\arcsin(2/r^2)$ depending on the radius, so that the origin, $p$ and $f(p)$ always span an isosceles triangle of area $1$; Lebesgue's density theorem then forces both $p$ and $f(p)$ into the set. Right triangles use the midpoint of $p$ and $f(p)$ in place of $f(p)$, and trapezoids come from cutting the apex off the isosceles triangle. The paper credits the approach of Kovač and Predojević as its inspiration. The source card lists the theorems and the sharpness remark.

Formalization. The site's label carries a Lean qualification. The formal-conjectures statements of the problem and its variants are marked solved and point to one Lean 4 file in the repository Jayyhk/erdos-lean, linked above at a pinned commit, which declares itself a formalization of Theorems 1 and 2 of this paper (and, in a second part, of Kovač and Predojević's results) and names no AI system. This corpus has not built or audited that development, so it is a link on this page and not evidence of acceptance.

Acceptance. The paper is refereed: J. Koizumi, Isosceles trapezoids of unit area with vertices in sets of infinite planar measure, Proc. Amer. Math. Soc. 153 (2025), no. 11, 4753–4758, published electronically 2025-08-29, DOI 10.1090/proc/17322. The curator of erdosproblems.com, Thomas Bloom, labels the problem proved and credits this paper with resolving the question, naming the trapezoid and the two triangles.