Rational Triangles

A rational triangle can be defined as one having all sides with rational length.

Rational Right Triangles - Early Solutions

In Śulba solution for the equation $x^{2}+y^{2}=z^{2}$-------(1) is available. Baudhāyana (c. 800 B.C.), Āpastamba and Kātyāyana (c. 500 B.C.) gave a method for the transformation of a rectangle into a square, which is the equivalent of the algebraical identity.

${\displaystyle mn=\left(m-{\frac {m-n}{2}}\right)^{2}-\left({\frac {m-n}{2}}\right)^{2}}$

where m, n are any two arbitrary numbers. Thus we get

${\displaystyle =({\sqrt {mn}})^{2}+\left({\frac {m-n}{2}}\right)^{2}=\left({\frac {m+n}{2}}\right)^{2}}$

substituting p²,q² for m, n respectively in order to eliminate the irrational quantities, we get

${\displaystyle =p^{2}q^{2}+\left({\frac {p^{2}-q^{2}}{2}}\right)^{2}=\left({\frac {p^{2}+q^{2}}{2}}\right)^{2}}$

which gives the rational solution of (1).

Kātyāyana gives a very simple method for finding a square equal to the sum of a number of other squares of the same size which gives us with another solution of the rational right triangle.

Kātyāyana says: "As many squares (of equal size) as you wish to combine into one, the transverse line will be (equal to) one less than that; twice a side will be (equal to) one more than that; (thus) form (an isosceles) triangle. Its arrow (i.e., altitude) will do that."

<Triangle picture to be added here>

For combining n squares of sides a each we form the isosceles triangle ABC such that $AB=AC={\frac {(n+1)a}{2}}$ and $BC=(n-1)a$

Then $AD^{2}=na^{2}$ which gives the formula

${\displaystyle =a^{2}({\sqrt {n}})^{2}+a^{2}\left({\frac {n-1}{2}}\right)^{2}=a^{2}\left({\frac {n+1}{2}}\right)^{2}}$

put m² for n in order to make the sides of the right angled triangle without the radical, we have

${\displaystyle =m^{2}a^{2}+a^{2}\left({\frac {m^{2}-1}{2}}\right)^{2}=a^{2}\left({\frac {m^{2}+1}{2}}\right)^{2}}$ which gives the rational solution of (1).

Mahāvīra's Definitions

Mahāvīra says A triangle or a quadrilateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called janya which means generated, formed or that which is generated or formed. Numbers which are involved in forming a particular figure are called its bīja-saṃkhyā (element-numbers) or simply bīja (element or seed).