Heronian Triangle Generator

A Heronian triangle is one whose side lengths and area are integers. They are named after the Greek mathematician Heron of Alexandria, whose name is also attached to the formula for triangular area in terms of side lengths (Heron's Formula).

Pythagorean triangles (right triangles whose sides are all integers) are the most prominent example of Heronian triangles, but they are not the only kind. For example, if you take two Pythagorean triangles that have a common leg length and put them together, you will create a new triangle that has integers side lengths and area. In fact, every Heronian triangle can be broken into two right triangles whose sides and areas are rational numbers.

Just as you can generate Pythagorean triangles with a two-parameter generator system, you can also find Heronian triangles with a three-parameter generator set:

Let m, n, and k be positive integers, and let

x = n(m² + k²)
y = m(n² + k²)
z = (m + n)|mn - k²|

Then x, y, and z form the sides of a Heronian triangle whose area is given by the formula

Area = mnk(m + n)|mn - k²|

The generating formula above will not generate all primitive Heronian triangles (for example, there are no values of m, n, and k that will generate the Heronian triangle with sides 5, 12, and 13), but it will generate at least one multiple of every primitive Heronian triangle. For example, if you choose m = 2, n = 3, and k = 2, you generate the triangle with sides 10, 24, and 26.

The values of m and n can be switched with no effect on the generated triangle, but if you exchange m and k, or n and k, a different Heronian triangle will be generated. For instance, choosing m = 2, n = 2, and k = 3 yields a Heronian triangle with sides 10, 26, and 26.

Here is a table of primitive Heronian triangles ordered by increasing area. See also Nearly Equilateral Heronian Triangles.

Shortest Side		Longest Side	Perimeter	Area
3	4	5	12	6
5	5	6	16	12
5	5	8	18	12
4	13	15	32	24
5	12	13	30	30
9	10	17	36	36
3	25	26	54	36
7	15	20	42	42
10	13	13	36	60
8	15	17	40	60
13	13	24	50	60
6	25	29	60	60
11	13	20	44	66
5	29	30	64	72
13	14	15	42	84
10	17	21	48	84
7	24	25	56	84
8	29	35	72	84
12	17	25	54	90
4	51	53	108	90
19	20	37	76	114
16	17	17	50	120
17	17	30	64	120
16	25	39	80	120
13	20	21	54	126
15	28	41	84	126
5	51	52	108	126
11	25	30	66	132
15	26	37	78	156
13	40	51	104	156
14	25	25	64	168
10	35	39	84	168
25	25	48	98	168
13	30	37	80	180
9	40	41	90	180
12	55	65	132	198
17	25	26	68	204
20	21	29	70	210
17	25	28	70	210
17	28	39	84	210
12	35	37	84	210
7	65	68	140	210
3	148	149	300	210
9	73	80	162	216
15	41	52	108	234
13	37	40	90	240
15	34	35	84	252
13	40	45	98	252
9	65	70	144	252
15	37	44	96	264
33	34	65	132	264
27	29	52	108	270
17	65	80	162	288
25	51	74	150	300
5	122	123	250	300
20	37	51	108	306
17	39	44	100	330
25	33	52	110	330
11	60	61	132	330
11	100	109	220	330
17	40	41	98	336
24	35	53	112	336
15	52	61	128	336
4	193	195	392	336
25	29	36	90	360
18	41	41	100	360
41	41	80	162	360
13	68	75	156	390
34	55	87	176	396
11	90	97	198	396
13	109	120	242	396
20	51	65	136	408
24	37	37	98	420
25	34	39	98	420
29	29	40	98	420
29	29	42	100	420
21	41	50	112	420
26	35	51	112	420
25	39	56	120	420
14	61	65	120	420
37	37	70	144	420
26	51	73	150	420
41	50	89	180	420
26	73	97	196	420
21	85	104	210	420
15	106	119	240	420
7	169	174	350	420

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