Heronian Triangles, A Wonderful Number Theory Topic for Middle School, High School, and Beyond

For this post, I will consider the definition of "Heronian Triangle" to be a triangle with integer sides and integer area. The common alternative of rational units for sides and area seems to be too broad for my taste for use in a middle school setting, although you may extend the definitions as you wish for your class, after all, it is your class.If you introduce the Heronian formula for the area of a triangle, this is a wonderful way to provide practice in using the formula.  \( A^2 = {s(s-a)(s-b)(s-c)}\) where \(s= \frac{a+b+c}{2} \)  And if you need more reason to instruct your students on this topic, consider that, among others, Euler and Gauss both worked on this topic. 

I imagine the easiest introduction is to simply ask, "Is it possible for a triangle to have all integer sides and integer area?"
Students familiar with Pythagorean triples will almost surely discover that some of the ones they know best, actually meet the definition, for example the student favorite 3-4-5 triangle has area of 6. You might mention how incredible that the numbers are in sequence. If a student comes up with the 5-12-13 right triangle, Point out that the area and perimeter are the same integer, 30. Are there any other Heronian triangles that have the same number for area and perimeter? Such triangles are usually called equable. If they don't get this one you can mention it later. In fact there are only five, challenge them to collect the whole set.

After they discover that they can produce an infinite number of Heronian Triangles from right Pythagorean triangles, we might ask if those are the only ones? Are there any non-Right triangles that are Heronian. Some questions you may want them to have time to think about over night. If someone is doing trial and error, they may come across the 5-5-6 triangle which has an area of 12, which means the altitude from the side of six is 4, and in fact,

it can be made of two 3-4-5 triangles. Could you make other non-right triangles by putting together two Pythagorean right triangles? For the teacher, it is good to point out that around 600 AD, BrahamaGupta, the great Indian mathematician and astronomer, pointed out that if you pick any three rational numbers, a, b, and c, then x= \( \frac{1}{2} (\frac{a^2}{b}+b) \) ; y= \( \frac{1}{2} (\frac{a^2}{c}+c) \) and z= \( \frac{1}{2} (\frac{a^2}{b}-b) \) + \( \frac{1}{2} (\frac{a^2}{c}-c) \) will form a scalene triangle which is rational in it's sides and height, and thus it's area. If appropriate values of a, b, and c are selected, these can form integer sides and area. BrahmaGupta also pointed out that in each case, the triangle is formed from two right triangles with a common side length for the altitude. 
Students, or teachers, who liked "The Hitchhiker's Guide to the Galaxy" may enjoy the 7-15-20 triangle which has both perimeter and area of 42. Author Douglas Adams set 42 as the answer to "life, the universe, and everything" and Tony Crilly and Colin Fletcher have dubbed this the "hitchhiker triangle."

Almost every case offers a topic for further exploration, the 3-4-5 triangle is special because it has all three sides in consecutive integers; can there be more of these? With one of two clever programmers in class, this is an easy one to produce a short list, but can they discover a function to predict them. In fact there is a somewhat easy generating function. If we have one that is n-1, n, and n+1, then we can find the next n by the simple Lucas Function \( n_{t+1} = 4 n_t - n_{t-1} \) 
If they haven't found  any others, we might challenge them to defend a 1-2-3 triangle as Heronian.  Is the area an integer?
Using 4 of the 3-4-5 triangle as one middle term, and 2 as the previous middle term, then the function will predict 4(4)-2 as the middle term of another sequential Heronian triangle, 13, 14, 15.  In fact this one was discovered by Heron himself, and he pointed out that it had an area of 84.