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

264 BRAHMAGUPTA AS AN ALGEBRAIST
Thus the two sides of the rational scalene triangle are 15
and 13. The base is .
½ ( 12²/6 - 6 ) + ½ ( 12²/8 - 8 ) = 9 + 5 = 14
The altitude is m = 12; area is equal to (base × altitude) / 2
= (14 × 12) / 2 = and the segments are √(13² - 12²) = 5 and
√(15² - 12²) = 9. Thus they are all integers.
Rational Isosceles Trapeziums
Brahmagupta has given us a method of obtaining such
isosceles trapeziums whose sides, diagonals, altitude, segments
and area are all rational numbers. His rule is as follows :
The diagonals of the rec-
tangle (generated) are the flank
sides of an isosceles trapezium;
the square of its side is divided
by an optional number and
then lessened by that optional
number and divided by two;
(the result) increased by the
upright is the base and lessened
by it is the face.¹
Fig. 21
Here in the figure, we have the isosceles trapezium ABCD
of which C D is the base and A B is known as the fase. Accord-
ing to Brahmagupta's rule, we have ( p being the optional
number).
CD = ½ ( (4m²n² - p) / p ) + ( m² - n² ) (base)
AB = ½ [ 4m²n²/p - p ] - [ m² - n² ] (face)
DH = ( m² - n² ) (upright)

1. आयतकर्णौ बाहू भुजकृतिरिष्टेन भाजितेष्टोना ।
   द्विहृता कोट्यधिका भूर्मुखमूना दिसमचतुरस्रे ॥ —BrSpSi. XII. 36