Brahmagupta triangle explained

A Brahmagupta triangle is a triangle whose side lengths are consecutive positive integers and area is a positive integer.^{[1] [2] [3]} The triangle whose side lengths are 3, 4, 5 is a Brahmagupta triangle and so also is the triangle whose side lengths are 13, 14, 15. The Brahmagupta triangle is a special case of the Heronian triangle which is a triangle whose side lengths and area are all positive integers but the side lengths need not necessarily be consecutive integers. A Brahmagupta triangle is called as such in honor of the Indian astronomer and mathematician Brahmagupta (c. 598 – c. 668 CE) who gave a list of the first eight such triangles without explaining the method by which he computed that list.^{[1] [4]}

A Brahmagupta triangle is also called a Fleenor-Heronian triangle in honor of Charles R. Fleenor who discussed the concept in a paper published in 1996.^{[5] [6] [7] [8]} Some of the other names by which Brahmagupta triangles are known are super-Heronian triangle^[9] and almost-equilateral Heronian triangle.^[10]

The problem of finding all Brahmagupta triangles is an old problem. A closed form solution of the problem was found by Reinhold Hoppe in 1880.^[11]

Generating Brahmagupta triangles

Let the side lengths of a Brahmagupta triangle be

t -1 , t and t +1 where t is an integer greater than 1. Using Heron's formula, the area A of the triangle can be shown to beA = ( \tfrac{t}{ 2 } ) \sqrt{ 3 [ ( \tfrac{t}{ 2 } ) ^{2} -1 ] } Since A has to be an integer, t must be even and so it can be taken as t =2 x where x is an integer. Thus, A = x\sqrt{ 3 (x ^{2-1)} } Since \sqrt{ 3 (x ^{2-1)} } has to be an integer, one must have x ^{2-1} =3 y ^{2} for some integer y . Hence, x must satisfy the following Diophantine equation:x ^{2-3y} ^{2=1} .This is an example of the so-called Pell's equation x ^{2-Ny} ^{2=1} with N =3 . The methods for solving the Pell's equation can be applied to find values of the integers x and y .

Obviously

x =2 , y =1 is a solution of the equation x ^{2-3y} ^{2=1} . Taking this as an initial solution x _{1=2,} y _{1=1} the set of all solutions \{(x _{n,} y _{n)\}} of the equation can be generated using the following recurrence relations^[1] x _{n+1} =2 x _{n+3y} _{n,} y _{n+1} = x _{n+2y} _{n} for n =1 , 2 ,\ldots or by the following relations\begin{align} x _{n+1} & = 4 x _{n} - x _{n-1} for n =2 , 3 ,\ldots with x _{1=2,} x _{2=7\\ y} _{n+1} & = 4 y _{n} - y _{n-1} for n =2 , 3 ,\ldots with y _{1=1,} y _{2=4. \end{align}} They can also be generated using the following property:x _{n+\sqrt{3}} y _{n=(x} _{1+\sqrt{3}y} ^{n for} ^{1 )} n =1 , 2 , \ldots The following are the first eight values of x _{n} and y _{n} and the corresponding Brahmagupta triangles:

n	1	2	3	4	5	6	7	8
x _{n}	2	7	26	97	362	1351	5042	18817
y _{n}	1	4	15	56	209	780	2911	10864
Brahmagupta triangle	3,4,5	13,14,15	51,52,53	193,194,195	723,724,725	2701,2702,2703	10083,10084,10085	37633,37634,337635

The sequence \{x _{n\}} is entry in the Online Encyclopedia of Integer Sequences (OEIS) and the sequence \{y _{n\}} is entry in OEIS.

Generalized Brahmagupta triangles

In a Brahmagupta triangle the side lengths form an integer arithmetic progression with a common difference 1. A generalized Brahmagupta triangle is a Heronian triangle in which the side lengths form an arithmetic progression of positive integers. Generalized Brahmagupta triangles can be easily constructed from Brahmagupta triangles. If

t -1 , t, t +1 are the side lengths of a Brahmagupta triangle then, for any positive integer k , the integers k(t -1 ), kt, k(t +1 ) are the side lengths of a generalized Brahmagupta triangle which form an arithmetic progression with common difference k . There are generalized Brahmagupta triangles which are not generated this way. A primitive generalized Brahmagupta triangle is a generalized Brahmagupta triangle in which the side lengths have no common factor other than 1.^[12]

To find the side lengths of such triangles, let the side lengths be

t - d, t, t + d where b,d are integers satisfying 1 \le d\le t . Using Heron's formula, the area A of the triangle can be shown to be A = ( \tfrac{b}{ 4 } ) \sqrt{ 3 (t ^{2-4d} ^{2)}} .For A to be an integer, t must be even and one may take t =2 x for some integer. This makesA = x\sqrt{ 3 (x ^{2-d} ^{2)}} .Since, again, A has to be an integer, x ^{2-d} ^{2} has to be in the form 3 y ^{2} for some integer y . Thus, to find the side lengths of generalized Brahmagupta triangles, one has to find solutions to the following homogeneous quadratic Diophantine equation:x ^{2-3y} ^{2=d} ^{2} .It can be shown that all primitive solutions of this equation are given by^[12] \begin{align} d & = \vert m ^{2} - 3 n ^{2\vert} /g\\ x & = (m ^{2} + 3 n ^{2)/g\\ y} & = 2 mn/g \end{align} where m and n are relatively prime positive integers and g = gcd (m ^{2} - 3 n ^{2,} 2 mn, m ^{2} + 3 n ^{2)} .

If we take

m = n =1 we get the Brahmagupta triangle ( 3 , 4 , 5 ) . If we take m =2 , n =1 we get the Brahmagupta triangle ( 13 , 14 , 15 ) . But if we take m =1 , n =2 we get the generalized Brahmagupta triangle ( 15 , 26 , 37 ) which cannot be reduced to a Brahmagupta triangle.

See also

• Brahmagupta polynomials
• Brahmagupta quadrilateral

Notes and References

1. R. A. Beauregard and E. R. Suryanarayan . The Brahmagupta Triangles . The College Mathematics Journal . January 1998 . 29 . 1 . 13-17 . 6 June 2024.
2. Web site: G. Jacob Martens . Rational right triangles and the Congruent Number Problem . arxiv.org . Cornell University . 6 June 2024.
3. Herb Bailey and William Gosnell . Heronian Triangles with Sides in Arithmetic Progression: An Inradius Perspective . Mathematics Magazine . October 2012 . 85 . 4 . 290-294 . 10.4169/math.mag.85.4.290.
4. Book: Venkatachaliyengar, K. . 1988 . Subbarayappa . B. V. . Scientific Heritage of India: Proceedings of a National Seminar, September 19-21, 1986, Bangalore . The Mythic Society, Bangalore . 36-48 . The Development of Mathematics in Ancient India: The Role of Brahmagupta .
5. Charles R. Fleenor . Heronian Triangles with Consecutive Integer Sides . Journal of Recreational Mathematics . 1996 . 28 . 2 . 113-115.
6. Web site: N. J. A. Sloane . A003500 . Online Encyclopedia of Integer Sequences . The OEIS Foundation Inc. . 6 June 2024.
7. Web site: Definition:Fleenor-Heronian Triangle . Proof-Wiki . 6 June 2024.
8. Vo Dong To . Finding all Fleenor-Heronian triangles . Journal of Recreational Mathematics . 2003 . 32 . 4 . 298-301.
9. Web site: William H. Richardson . Super-Heronian Triangles . www.wichita.edu . Wichita State University . 7 June 2024.
10. Roger B Nelsen . Almost Equilateral Heronian Triangles . Mathematics Magazine . 2020 . 93 . 5 . 378-379.
11. H. W. Gould . A triangle with integral sides and area . Fibonacci Quarterly . 1973 . 11 . 27-39 . 7 June 2024.
12. James A. Macdougall . Heron Triangles With Sides in Arithmetic Progression . Journal of Recreational Mathematics . January 2003 . 31 . 189-196.