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Concept
Heronian triangle
Summary
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. 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.
Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the Diophantine equation
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.
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; 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.
Scaling a triangle with a factor of s consists of multiplying its side lengths by s; this multiplies the area by and produces a similar triangle. Scaling a rational Heronian triangle by a rational factor produces another rational Heronian triangle.
Given a rational Heronian triangle of side lengths the scale factor produce a rational Heronian triangle such that its side lengths are setwise coprime integers. It is proved below that the area A is an integer, and thus the triangle is a Heronian triangle. Such a triangle is often called a primitive Heronian triangle.
In summary, every similarity class of rational Heronian triangles contains exactly one primitive Heronian triangle.
Official source
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