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<p>The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century. Despite its age, it would have been accessible to Euclid and his contemporaries. The theorem remains evergreen, with new proofs continuing to appear steadily. The theorem has fostered discussion about the nature of proof itself, direct and indirect. Here we continue the momentum by providing a trigonometric proof, relatively short, based on an analytic estimate that leverages algebraic trigonometric identities. Many proofs of the theorem exist in the literature. Some of these contain key ideas that already appeared in C.L. Lehmus’ 1850 proofs, not always with citation. In the aim of increasing awareness of and making more accessible Lehmus’ proofs, we provide an annotated translation. We conclude with remarks on different proofs and relations among them.</p>

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Echos of the Steiner-Lehmus equal bisectors theorem

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Christoph Börgers,
  • Eric L. Grinberg,
  • Mehmet Orhon,
  • Junhao Shen

摘要

The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century. Despite its age, it would have been accessible to Euclid and his contemporaries. The theorem remains evergreen, with new proofs continuing to appear steadily. The theorem has fostered discussion about the nature of proof itself, direct and indirect. Here we continue the momentum by providing a trigonometric proof, relatively short, based on an analytic estimate that leverages algebraic trigonometric identities. Many proofs of the theorem exist in the literature. Some of these contain key ideas that already appeared in C.L. Lehmus’ 1850 proofs, not always with citation. In the aim of increasing awareness of and making more accessible Lehmus’ proofs, we provide an annotated translation. We conclude with remarks on different proofs and relations among them.