In all the (Euclidean) world, up to simple scaling, there is only one pair of triangles with the following properties:
- One triangle is a right triangle and one is isosceles,
- All side lengths of both triangles are rational numbers, and
- The perimeters and areas of both triangles are equal.
The right triangle of this pair has side lengths (135, 352, 377), and the isosceles has side lengths (132, 366, 366). If you’re suspicious, you can easily add up the side lengths to see that their perimeters are the same. Computing areas is a bit trickier. It’s simple to compute the area of a right triangle knowing only its side lengths: it is half of the product of the shorter two sides. For the isosceles triangle, you can either apply Heron’s formula, which gives you the area of a triangle using only its side lengths, or first find the altitude of the triangle using the Pythagorean theorem and then figure out the area of the triangle from there. Either way you do it, you will find that each triangle’s perimeter is 864 units and area is 23,760 square units.
I had never thought about trying to find two rational triangles with the same perimeters and areas before, so I didn’t know how to feel when I found out. (Why I was so worried about knowing how to feel is an interesting question and beyond the scope of this blog post.) Is this surprising? If so, is it surprising that there is only one such pair, or surprising that there aren’t more? Should I be surprised that the side lengths of these very special triangles are as large as they are, or as small? Should I be surprised that the paper showing that this pair is unique came out just last year? Or that it is only five pages long? Is nothing a surprise at all, or is everything surprising? This answer to a question I never knew I had left me with even more questions.
The proof that (135, 352, 377) and (132, 366, 366) form the unique pair of triangles with the desired properties comes from a field of math called algebraic geometry. To oversimplify a bit, algebraic geometry is like your high school algebra classes—understanding relationships between symbolic equations and geometric figures in a plane or higher-dimensional space—turned up a notch. A central question in much of algebraic geometry is how to determine whether a given equation has any solutions that are integers or rational numbers, and if so, how many. (For a solution to count as rational, all the variables must take rational values. That is, if the equation has two variables, x and y, a rational solution would be one in which both x and y are rational numbers.) For example, the equation x2−y2=5 has infinitely many rational solutions and a few integer solutions, but the equation x3−y3=5 has only finitely many rational and no integer solutions. It makes sense that rational points are hard to find: there are a lot more irrational numbers than rational numbers, after all. But some polynomials have a lot of rational solutions, and some have none.
Yoshinosuke Hirakawa and Hideki Matsumura, the authors of the paper revealing the unique pair of triangles, show that finding such a pair is equivalent to finding rational solutions of a particular equation. Then they invoke some theorems about how many rational solutions an equation with certain properties can have, chase some leads for potential rational solutions, and find that only one actually gives them valid triangles. The proof is short but requires some high-powered tools.
All the side lengths of the isosceles triangle in the special pair are even. Hirakawa and Matsumura include an appendix that shows that if we ask for both triangles to be primitive—that is, each triangle’s side lengths are all integers, and they have no common factors larger than 1—no pair of triangles will satisfy all three criteria. The proof that there is no primitive pair satisfying all the requirements is quite a bit simpler than the proof that their special pair is unique. On the other hand, without requiring the triangles to be right or isosceles, there are infinitely many pairs of rational triangles that have the same perimeters and areas. I haven’t fully resolved how to feel about it, but I think the problem is one more example of the fact that in mathematics, the boundaries between finite and infinite, and easy and hard, can be delicate and surprising in and of themselves.