Problem 97

Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?

Geometry
Distances
Convexity

Wiki page

Status

ClaimedDisproved

A full solution has been claimed but not yet accepted. The statement is false.

Falsifiable on the site: its export of 2026-09-04 records the label "FALSIFIABLE", and its page was last edited 27 October 2025; its proof-claims tab carries one full proof claim, the manuscript of Liam Kruer, Jensen Kohlmeyer and Liam Price of 13 September 2026, recorded on its claim page (Kruer Kohlmeyer Price, 2026) without being adopted; the standing in the frontmatter is derived from the claim pages.

Er46b: Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248-250.
Er75f: Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
Er87b: Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
FiRe92: Fishburn, P. C. and Reeds, J. A., Unit distances between vertices of a convex polygon. Comput. Geom. (1992), 81-91.

Problem 97

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Proof claims

Discussion