Heronian Triangle

A Triangle with Rational side lengths and Rational Area. Brahmagupta gave a parametric solution for integer Heronian triangles (the three side lengths and area can be multiplied by their Least Common Multiple to make them all Integers): side lengths $c(a^2+b^2)$, $b(a^2+c^2)$, and $(b+c)(a^2-bc)$, giving Semiperimeter

$$s=a^2(b+c)$$

and Area

$$\Delta=abc(a+b)(a^2-bc).$$

The first few integer Heronian triangles, sorted by increasing maximal side lengths, are (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (4, 13, 15), (13, 14, 15), (9, 10, 17), ... (Sloane's A046128, A046129, and A046130), having areas 6, 24, 30, 54, 24, 84, 36, ... (Sloane's A046131).

Schubert (1905) claimed that Heronian triangles with two rational Medians do not exist (Dickson 1952). This was shown to be incorrect by Buchholz and Rathbun (1997), who discovered six such triangles.

See also Heron's Formula, Median (Triangle), Pythagorean Triple, Triangle

References

Buchholz, R. H. On Triangles with Rational Altitudes, Angle Bisectors or Medians. Doctoral Dissertation. Newcastle, England: Newcastle University, 1989.

Buchholz, R. H. and Rathbun, R. L. ``An Infinite Set of Heron Triangles with Two Rational Medians.'' Amer. Math. Monthly 104, 107-115, 1997.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 199 and 208, 1952.

Guy, R. K. ``Simplexes with Rational Contents.'' §D22 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 190-192, 1994.

Kraitchik, M. ``Heronian Triangles.'' §4.13 in Mathematical Recreations. New York: W. W. Norton, pp. 104-108, 1942.

Schubert, H. ``Die Ganzzahligkeit in der algebraischen Geometrie.'' In Festgabe 48 Versammlung d. Philologen und Schulmänner zu Hamburg. Leipzig, Germany, pp. 1-16, 1905.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Puzzles. London: Penguin Books, p. 34, 1992.