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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,

  1. is EVERY rational number the area of some GENERAL rational triangle?

  2. 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

Nandakumar R
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1 Answers1

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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.

lulu
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  • That's right,... but even many non-perfect square rationals (indeed integers) such as 5, 6, 7,... are the areas of right rational triangles (pls see wiki article on congruent numbers). So we are looking for a wider answer. Thanks! – Nandakumar R Jul 02 '19 at 10:13
  • Sure. For the case of a general integer area, see this – lulu Jul 02 '19 at 10:20