Let us define a 'rational triangle' as one with lengths of all sides rational.
We are aware that a positive integer is called 'congruent' only if it is the area of a RIGHT triangle with rational length sides; so we have, every rational number is NOT the area of some 'right rational triangle'. However,
is EVERY rational number the area of some GENERAL rational triangle?
for two given rational numbers A and P, among the infinitely many general triangles with area A and perimeter P (there are infinitely many such triangles if A and P are within a suitable range), is there a guarantee that there are any (or infinitely many) rational triangles?
with regards,
R. Nandakumar,
K. Shesadri
As a start: The triangle with sides $\left( \frac 32, \frac 53, \frac {17}6\right)$ has area $1$, by Heron's formula. Thus any rational number which is a perfect square is the area of a rational triangle.