An Alternative Proof of Ptolemy’s Theorem and its Variations

This paper introduces a pure geometric proof for Ptolemy’s Theorem, without using trigonometry, coordinate geometry, complex numbers, vectors or any other geometric inversion techniques focusing on cyclic quadrilaterals and employing a generalized identity in relation to a cevian of an arbitrary Euclidean plane triangle. Additionally, the paper provides proofs to the converse of Ptolemy’s Theorem to which almost no pure geometric complete proof is available, and to the standard Ptolemy’s Inequality, to fulfil the research gap in the proofs to some extent. It also includes applications, new corollaries, derived from Ptolemy’s Theorem and its converse.