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<p>We investigate the question of whether a simple closed curve in the plane must contain all four vertices of some rhombus having one diagonal collinear with a specified point. This complements previous research on whether there is a rhombus with a diagonal or side parallel to a given line. We obtain a new proof that a simple closed curve contains the vertices of uncountably many rhombi. We also explore conditions guaranteeing that all points in some region are collinear with diagonals of such rhombi.</p>

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Inscribed rhombi having diagonals collinear with specified points

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  • 发表日期：

    2026年6月3日

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  • Stephen E. Wright

摘要

We investigate the question of whether a simple closed curve in the plane must contain all four vertices of some rhombus having one diagonal collinear with a specified point. This complements previous research on whether there is a rhombus with a diagonal or side parallel to a given line. We obtain a new proof that a simple closed curve contains the vertices of uncountably many rhombi. We also explore conditions guaranteeing that all points in some region are collinear with diagonals of such rhombi.