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

ISOSCELES TRIANGLES 263 rational isosceles triangle with altitude 8 are (10,10, 12). Rational Scalene Triangles: Brahmagupta lays down the following rule in the case of rational scalene triangle : The square of an optional number is divided twice by two arbitrary numbers; the moieties of the sums of the quotients and (respective) optional numbers are the sides of a scalene triangle; the sum of the moieties of the differences is the base.¹ In other words, if m, p, q are any rational numbers, then the sides of a rational scalene triangle are : ½ ( m²/p + p ), ½ ( m²/q + q ), ½ ( m²/p - p ) + ½ ( m²/q - q ) Here the altitude (m), area and segments of the base of this triangle are all rational. Thus putting m=12, p=6, and q=8 in Brahmagupta's gene- ral equation, Pṛthūdaka Svāmī derives a scalene triangle with sides (13,15) and (14) altitude (12), area (84 and the segments of the base (5) which are all integral numbers. ½ ( m²/p + p ) = ½ ( 12²/6 + 6 ) = 15; ½ ( m²/q + q ) = ½ ( 12²/8 + 8 ) = 13 B' A C' A' B' C' B H C H B C Fig. 19 Fig. 20

1. इष्टद्वयेन भक्तो द्विधेष्ट वर्ग फलेष्टयोगार्धे । विषमत्रिभुजस्य भुजाविष्टोनफलार्धयोगो भूः ॥ —BrSpSi. XII, 34.