Heronian triangle

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Heronian triangles are triangles whose three side lengths a, b, c and their area A are all integers. The classic example is a triangle with sides 13, 14, 15, which has area 84. These triangles are connected to Heron’s formula, which gives the area from the side lengths. A Heronian triangle is one where that area comes out as an integer.

Primitive and rational versions
- Primitive: If the greatest common divisor of the three side lengths is 1, the triangle is primitive.
- Rational Heronian: If the side lengths and the area are all rational numbers, the triangle is a rational Heronian triangle. Every integral Heronian triangle is rational, and every rational Heronian triangle is similar to exactly one primitive Heronian triangle.

What else is special
- In a rational Heronian triangle, many related quantities are also rational, such as the altitudes, the circumradius, the inradius, the exradii, and the sines and cosines of the angles.
- You can get more Heronian triangles by scaling: multiplying all sides by a rational number produces another rational Heronian triangle. One of the main ideas is that every similarity class of rational Heronian triangles has a unique primitive member.

Ways to generate Heronian triangles
- Brahmagupta’s method and Euler’s method are two famous ways to produce Heronian triangles. They use parameters (numbers you choose) to build the side lengths and the area. Neither method always gives a primitive triangle, but they can generate all Heronian triangles in the right way.
- A simple, concrete example: start with two Pythagorean triples, such as 3-4-5 and its mirror image. Joining them along the leg of length 3 gives a Heronian triangle with sides 5, 5, 6 and area 12. Another example from this idea is a non‑right Heronian triangle with sides 5, 29, 30 and area 72.
- A note: not all Heronian triangles come from joining Pythagorean triangles, but every Heronian triangle is similar to one that can be built this way or by the standard parameter formulas.

A few fun facts
- There are infinitely many Heronian triangles, and many different families of them exist.
- There are exactly five equable Heronian triangles—triangles whose area equals their perimeter. They are (5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17). Most of these are not primitive.
- Some Heronian triangles have interesting special properties, such as having all sides that are perfect squares or being able to place the triangle on a lattice in special ways. These connections link Heronian triangles to other problems in number theory and geometry.

In short, Heronian triangles are a beautiful intersection of integers and geometry: triangles with integer sides and integer areas, with rich ways to construct and classify them, and with many interesting special cases and applications.