The New Proof of Ptolemy's Theorem & Nine Point Circle Theorem

摘要

The main purpose of the paper is to present a new proof of the two celebrated theorems: one is "Ptolemy's Theorem" which explains the relation between the sides and diagonals of a cyclic quadrilateral and another is "Nine Point Circle Theorem" which states that in any arbitrary triangle the three midpoints of the sides, the three feet of altitudes, the three midpoints of line segments formed by joining the vertices and Orthocenter, total nine points are concyclic. Our new proof is based on a metric relation of circumcenter.