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 ...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.展开更多
Ellipses can be constructed by folding disks. These folds are forming an envelope of tangents to the ellipse. In the paper of Gorkin and Shaffer, it was shown that the ellipse constructed by folding can be circumscrib...Ellipses can be constructed by folding disks. These folds are forming an envelope of tangents to the ellipse. In the paper of Gorkin and Shaffer, it was shown that the ellipse constructed by folding can be circumscribed by an arbitrary triangle of tangents, the vertices of which are lying on the circumference of the disk. They offered two non-elementary methods of proof, one using Poncelet’s Theorem, the other employing Blaschke products. In this paper, it is the intention to present an elementary proof by means of analytic geometry.展开更多
文摘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.
文摘Ellipses can be constructed by folding disks. These folds are forming an envelope of tangents to the ellipse. In the paper of Gorkin and Shaffer, it was shown that the ellipse constructed by folding can be circumscribed by an arbitrary triangle of tangents, the vertices of which are lying on the circumference of the disk. They offered two non-elementary methods of proof, one using Poncelet’s Theorem, the other employing Blaschke products. In this paper, it is the intention to present an elementary proof by means of analytic geometry.
