Heronian triangle

Key takeaways

Heronian triangle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Heronian triangle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Heronian triangle from memory before moving on to harder problems.

Reference excerpt

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

16 A 2 = ( a + b + c ) ( a + b − c ) ( b + c − a ) ( c + a − b ) ; {\displaystyle 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. 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 geometrically 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 to primitive triangles Scaling a triangle with a factor of s consists of multiplying its side lengths by s; this multiplies the area by s 2 {\displaystyle s^{2}} 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 p d , q d , r d , {\textstyle {\frac {p}{d}},{\frac {q}{d}},{\frac {r}{d}},} the scale factor d gcd ( p , q , r ) {\textstyle {\frac {d}{\gcd(p,q,r)}}} produces a rational Heronian triangle such that its side lengths a , b , c {\textstyle a,b,c} 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. A byproduct of the proof is that exactly one of the side lengths of a primitive Heronian triangle is an even integer. Proof: One has to prove that, if the side lengths a , b , c {\textstyle a,b,c} of a rational Heronian triangle are coprime integers, then the area A is also an integer and exactly one of the side lengths is even. The Diophantine equation given in the introduction shows immediately that 16 A 2 {\displaystyle 16A^{2}} is an integer. Its square root 4 A {\displaystyle 4A} is also an integer, since the square root of an integer is either an integer or an irrational number. If exactly one of the side lengths is even, all the factors in the right-hand side of the equation are even, and, by dividing the equation by 16, one gets that A 2 {\displaystyle A^{2}} and A {\displaystyle A} are integers. As the side lengths are supposed to be coprime, one is left with the case where one or three side lengths are odd. Supposing that c is odd, the right-hand side of the Diophantine equation can be rewritten

( ( a + b ) 2 − c 2 ) ( c 2 − ( a − b ) 2 ) , {\displaystyle ((a+b)^{2}-c^{2})(c^{2}-(a-b)^{2}),}

with a + b {\displaystyle a+b} and a − b {\displaystyle a-b} even. As the square of an odd integer is congruent to 1 {\displaystyle 1} modulo 4, the right-hand side of the equation must be congruent to − 1 {\displaystyle -1} modulo 4. It is thus impossible, that one has a solution of the Diophantine equation, since 16 A 2 {\displaystyle 16A^{2}} must be the square of an integer, and the square of an integer is congruent to 0 or 1 modulo 4.

Examples Any Pythagorean triangle is a Heronian triangle. The side lengths of such a triangle are integers, by definition. In any such triangle, one of the two shorter sides has even length, so the area (the product of these two sides, divided by two) is also an integer.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Heronian triangle

Start with the simplest possible case. Write down what Heronian triangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Heronian triangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Heronian triangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Heronian triangle

In research: Heronian triangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Heronian triangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Heronian triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic problems of plane geometry, Triangles named after people, Types of triangles, so understanding it makes those chapters shorter.
In everyday life: Look for Heronian triangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Heronian triangle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Heronian triangle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Heronian triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heronian triangle in simple terms?

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…

Why does Heronian triangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Heronian triangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Heronian triangle.