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).