Summary
Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in , via a modified matrix version (reflection transformation) of a classical algebraic identity.
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Acknowledgments
The authors would like to thank the anonymous referee for particularly careful reading and constructive comments, as well as the Editor’s polishing in improving this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 We would be grateful to any reader who may know of the original reference for this complex-number based proof. As far as we know, although with minor differences in statements for later editions, Hardy already includes this proof as an example in the first (1908) edition of his famous treatise [p.101, 9].
Additional information
Notes on contributors
Jianke Chen
Jianke Chen is a lecturer in the faculty of data science and media intelligence at the Communication University of China in Beijing, China. He received his Ph.D. in mathematics in 2014. His interests include algebra, the popularization of mathematics, and badminton.
Kailiang Lin
Kailiang Lin is a lecturer in the college of science at Northwest A&F University in Shaanxi, China. He received his Ph.D. in mathematics in 2014. His interests include algebra, mathematics education, the popularization of mathematics, and volleyball.