Brahmagupta triangle
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] Let the side lengths of a Brahmagupta triangle be
Using Heron's formula, the area
must satisfy the following Diophantine equation: This is an example of the so-called Pell's equation
The methods for solving the Pell's equation can be applied to find values of the integers
Taking this as an initial solution
of the equation can be generated using the following recurrence relations[1] or by the following relations They can also be generated using the following property: The following are the first eight values of
and the corresponding Brahmagupta triangles: The sequence
is entry A001075 in the Online Encyclopedia of Integer Sequences (OEIS) and the sequence
is entry A001353 in OEIS.
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.
are the side lengths of a Brahmagupta triangle then, for any positive integer
are the side lengths of a generalized Brahmagupta triangle which form an arithmetic progression with common difference
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
Using Heron's formula, the area
Thus, to find the side lengths of generalized Brahmagupta triangles, one has to find solutions to the following homogeneous quadratic Diophantine equation: It can be shown that all primitive solutions of this equation are given by[12] where
are relatively prime positive integers and
we get the generalized Brahmagupta triangle
which cannot be reduced to a Brahmagupta triangle.
