ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

RATIONAL INSCRIBED QUADRILATERALS 267 circumscribed circles will be expressible in integers. Such quadrilaterals we shall call as Brahmagupta Quadrilaterals. The solution of this formidable problem has been given by Brahmagupta as follows : The upright and bases of two right-angled triangles being reciprocally multiplied by the diagonals of the other will give the sides of a quadrilateral of unequal sides : ( of these ) the greatest is the base, the least is the face, and the other two sides are the two flanks.¹ Taking Brahmagupta's integral solution, the sides of the two right triangles of reference are given by : (1) m²-n², 2 mn, m²+n²; (ii) p²-q², 2 pq, p²+q²; where m, n, p, q are integers. Then the sides of the Brah- magupta Quadrilateral are. (m²-n²)(p²+q²), (p²-q²)(m²+n²), 2mn(p²+q²), 2pq(m²+n²) (Arrangement A) Pṛthūdaka Svāmī has- illustrated the rational inscribed quadrilateral by taking an example of the right angle triangles. (i) (3,4,5) (m²-n²=3, m²+n²=5, whence m=2, n=1) (ii) (5,12,13)(p²-q²=5, p²+q²=13, whence p=3, q=2) Fig. 22 Substituting these values in the above equations, we get the sides of the quardilateral as ( 39, 25, 52 and 60).²

1. जात्यद्वय कोटिभुजाः परकर्णगुणा भुजाश्चतुर्विषमे । अधिको भूर्मुखंहीनो बाहुद्वितयं भुजावन्यौ ॥ —BrSpSi. XII. 38 2 The diagonals of this quadrilateral are given by Bhāskara II as 56 (=3.12+4.5) and 63 (=4.12+3.5). (Cont. on page 268)