Heronian triangle

In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths $a$ , $b$ , and $c$ and area $A$ are all positive integers.^[1]^[2] Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides $13, 14, 15$ and area $84$ .^[3]

Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the Diophantine equation

16\,A^{2}=(a+b+c)(a+b-c)(b+c-a)(c+a-b);

that is, the side lengths and area of any Heronian triangle satisfy the equation, and any positive integer solution of the equation describes a Heronian triangle.^[4]

If the three side lengths are setwise coprime (meaning that the greatest common divisor of all three sides is 1), the Heronian triangle is called primitive.

Triangles whose side lengths and areas are all rational numbers (positive rational solutions of the above equation) are sometimes also called Heronian triangles or rational triangles;^[5] in this article, these more general triangles will be called rational Heronian triangles. Every (integral) Heronian triangle is a rational Heronian triangle. Conversely, every rational Heronian triangle is similar to exactly one primitive Heronian triangle.

In any rational Heronian triangle, the three altitudes, the circumradius, the inradius and exradii, and the sines and cosines of the three angles are also all rational numbers.

^ Carlson, John R. (1970), "Determination of Heronian Triangles" (PDF), Fibonacci Quarterly, 8: 499–506
^ Beauregard, Raymond A.; Suryanarayan, E. R. (January 1998), "The Brahmagupta Triangles" (PDF), College Mathematics Journal, 29 (1): 13–17, doi:10.2307/2687630, JSTOR 2687630
^ Sastry, K. R. S. (2001). "Heron triangles: A Gergonne-Cevian-and-median perspective" (PDF). Forum Geometricorum. 1 (2001): 17–24.
^ The sides and area of any triangle satisfy the Diophantine equation obtained by squaring both sides of Heron's formula; see Heron's formula § Proofs. Conversely, consider a solution of the equation where $(a,b,c,A)$ are all positive integers. It corresponds to a valid triangle if and only if the triangle inequality is satisfied, that is, if the three integers $a+b-c,$ $b+c-a,$ and $c+a-b$ are all positive. This is necessarily true in this case: if any of these sums were negative or zero, the other two would be positive and the right-hand side of the equation would thus be negative or zero and could not possibly equal the left-hand side $16\,A^{2},$ which is positive.
^ Weisstein, Eric W. "Heronian Triangle". MathWorld.

[carlson-1] Carlson, John R. (1970), "Determination of Heronian Triangles" (PDF), Fibonacci Quarterly, 8: 499–506

[2] Beauregard, Raymond A.; Suryanarayan, E. R. (January 1998), "The Brahmagupta Triangles" (PDF), College Mathematics Journal, 29 (1): 13–17, doi:10.2307/2687630, JSTOR 2687630

[3] Sastry, K. R. S. (2001). "Heron triangles: A Gergonne-Cevian-and-median perspective" (PDF). Forum Geometricorum. 1 (2001): 17–24.

[4] The sides and area of any triangle satisfy the Diophantine equation obtained by squaring both sides of Heron's formula; see Heron's formula § Proofs. Conversely, consider a solution of the equation where $(a,b,c,A)$ are all positive integers. It corresponds to a valid triangle if and only if the triangle inequality is satisfied, that is, if the three integers $a+b-c,$ $b+c-a,$ and $c+a-b$ are all positive. This is necessarily true in this case: if any of these sums were negative or zero, the other two would be positive and the right-hand side of the equation would thus be negative or zero and could not possibly equal the left-hand side $16\,A^{2},$ which is positive.

[5] Weisstein, Eric W. "Heronian Triangle". MathWorld.

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