n Learning and Teaching Mathematics - An elementary proof of Ptolemy’s theorem

Volume 2019 Number 27
  • ISSN : 1990-6811



Ptolemy’s theorem is a well-known result in plane geometry and can be stated as follows:

If ABCD is a cyclic quadrilateral with sides a, b, c, d and diagonals e, f, then ac +bd = ef

In this article we present an elementary proof of Ptolemy’s theorem based on basic trigonometry, circle geometry and transformation geometry. The proof is different to those described elsewhere – see for example Johnson (1929), Coxeter & Greitzer (1967), Ostermann & Wanner (2012) and Miculita (2017).

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