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<p>The present paper contains further extensions of results given in [<CitationRef CitationID="CR1">1</CitationRef>], and includes another family of curves called <i>N</i>-power ellipses which are a generalization of the notions of ellipses, Cassini ovals and Fermat curves. We examine the chordal condition introduced in [<CitationRef CitationID="CR1">1</CitationRef>] of <i>N</i>-power ellipses at a fixed point, for natural and rational values of <i>N</i>. Next we formulate and prove the chordal condition for an ellipse, for an arbitrary point lying inside of it. The chordal condition is a generalization of a well-known property of ellipses, expressing relations of lengths from their foci to other points. This paper is based on Stewart’s theorem [<CitationRef AdditionalCitationIDS="CR6" CitationID="CR5">5</CitationRef>–<CitationRef CitationID="CR7">7</CitationRef>].</p>

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The chordal condition and Stewart’s theorem II

  • 期刊论文

  • 发表日期：

    2026年2月25日

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  • Waldemar Cieślak,
  • Witold Mozgawa,
  • Aharon Naiman

摘要

The present paper contains further extensions of results given in [1], and includes another family of curves called N-power ellipses which are a generalization of the notions of ellipses, Cassini ovals and Fermat curves. We examine the chordal condition introduced in [1] of N-power ellipses at a fixed point, for natural and rational values of N. Next we formulate and prove the chordal condition for an ellipse, for an arbitrary point lying inside of it. The chordal condition is a generalization of a well-known property of ellipses, expressing relations of lengths from their foci to other points. This paper is based on Stewart’s theorem [5–7].