It will be shown in this paper that if two chords of a closed plane convex curve θ divide θ into four arcs of equal length and intersect inside the domain bounded by θ, then the sum of the lengths of the two chords is at least equal to (√5 — 2)½ times the length of θ. We shall show firstly that we need only consider the case when θ is a convex (possibly degenerate) quadrilateral and then prove the result in this case.

This result is related to a conjecture of P. Ungar in which the chords are assumed to be perpendicular and the factor (√5 — 2)½ is replaced by §. But Ungar's conjecture is neither proved nor disproved by this result.