Heronian Triangle - Theorem

Theorem

Any Heronian triangle can be split into two right-angled triangles whose side lengths form Pythagorean triples with rational values.

Proof of the theorem

Consider again the illustration to the right, where it is known that c, e, b + d, and the triangle area A are integers. Assume that b + d is greater than or equal to c and e, so that the foot of the altitude perpendicular to this side falls within the side. To show that the triples (a, b, c) and (a, d, e) are rational Pythagorean triples, it suffices to show that a, b, and d are rational.

Since the triangle area is

one can solve for a to find

which is rational, as both and are integers. It remains to show that b and d are rational.

From the Pythagorean theorem applied to the two right-angled triangles, one has

and

One can subtract these two, to find

By assumption, c, e, and b + d are integers. So b − d is rational and hence

are both rational. Q.E.D.