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.