Construction and application of third diagonal in cyclic quadrilaterals by Nārāyaṇa Paṇḍita

In his comprehensive mathematical treatise Gaṇitakaumudī, Nārāyaṇa Paṇḍita has presented a nuanced, systematic, and elaborate exposition of cyclic quadrilaterals. Here, besides discussing its key properties, Nārāyaṇa fashions a “third diagonal" by interchanging two sides of a cyclic quadrilateral. He also provides a variety of mathematical expressions for computing the area, altitude, circumradius, and so on of a cyclic quadrilateral. It turns out that some of these expressions come out very elegant when we involve the third diagonal in them. In this paper, apart from bringing out the verses of Nārāyaṇa, we also present modern mathematical derivations for the results given by him pertaining to the cyclic quadrilateral.