An easy proof of Ptolemy’s theorem

Many proofs are available for the famous and important theorem in geometry known as Ptolemy’s theorem. For our discussion, we consider the proof presented by Shirali. In his article, he described a simple geometrical proof of the theorem and presented two elegant applications. He noted that the proof ‘presents a challenge’ because from the statement of the theorem we get no clue on how to tackle it. In the proof, there arises a crucial idea of locating a point E on a diagonal of the quadrilateral that enables the construction of two similar triangles. A recent demonstration by Tunsteno demonstrates a simple and intuitively appealing method for locating the point E. We present it here for the benefit of school students and teachers