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September 20, 2015

The simplest Heronian triangles

Filed under: mathematics — ckrao @ 12:05 pm

Heronian triangles are those whose side lengths and area have integer value. Most of the basic ones are formed either by right-angled triangles of integer sides, or by two such triangles joined together. Following the proof in [1] it is not difficult to show that such triangles have side lengths proportional to (x,y,z) = (n(m^2 + h^2), m(n^2 + h^2), (m+n)(mn-h^2)) where m,n and h are integers with mn > h^2.  Firstly, if a triangle has integer side lengths and area, its altitudes must be rational, being twice the area divided by a side length. Also by the cosine rule, the cosine of its angles must be rational, so z_1 and z_2 in the diagram below are rational too (here assume z is the longest side, so that the altitude is inside the triangle).

heronian_setupThis gives us the equations

\displaystyle h^2 = x^2 - z_1^2 = y^2 - z_2^2, z_1 + z_2 = z,\quad \quad (1)

where h, z_1, z_2 \in \mathbb{Q}. Letting x + z_1 = m and y + z_2 = n it follows from the above equations that x - z_1 = h^2/m, y-z^2 = h^2/n from which

\displaystyle (x,y,z) = \left(\left(\frac{1}{2}(m + \frac{h^2}{m}\right), \frac{1}{2}\left(n + \frac{h^2}{n}\right), \frac{1}{2}\left( m - \frac{h^2}{m} + (n - \frac{h^2}{n}\right)\right). \quad\quad (2)

Scaling the sides up by a factor of 2mn, the sides are proportional to

(x',y',z') = (n(m^2 + h^2), m(n^2 + h^2), (m+n)(mn-h^2)).\quad\quad(3)

Next, letting d be the common denominator of the rational numbers h, z_1 and z_2, we multiply the rational solution (x', y', z') in (3) each by d^3 to obtain an integral solution. The altitude upon side length z is proportional to 2hmn and the area is hmn(m+n)(mn-h^2). Hence if we start with positive m,n,h with no common factor and with mn > h^2, then (3) gives the side lengths of a Heronian triangle that can then be made primitive by dividing by a common factor.

Below the 20 primitive Heronian triangles with area less than 100 are illustrated to scale, where the first row has been doubled in size for easier viewing (a larger list is here). Note that all but one of them is either an integer right-angled triangle or decomposable into two such triangles as indicated by the blue numbers and sides. Refer to [2] for more on triangles which are not decomposable into two integer right-angled triangles. Here are the primitive Pythagorean triples that feature in the triangles:

  • 3-4-5
  • 5-12-13
  • 8-15-17
  • 20-21-29
  • 7-24-25
  • 28-45-53

heronian

References

[1] Carmichael, R. D., 1914, “Diophantine Analysis”, pp.11-13; in R. D. Carmichael, 1959, The Theory of Numbers and Diophantine Analysis, Dover Publications, Inc.

[2] Yiu, Paul (2008), Heron triangles which cannot be decomposed into two integer right triangles (PDF), 41st Meeting of Florida Section of Mathematical Association of America.

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