Unexpectedly Intriguing!
May 8, 2020

Can you draw two different triangles that meet each of the following rules?

  • 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.

It sounds like a pretty simple proposition, doesn't it? Right triangles are pretty well known commodities, and isosceles triangles aren't too far behind, since they are symmetric with two sides that are of equal length. The definitions of perimeter and area for triangles are very straightforward too.

What about rational numbers? Well, since those encompass the sets of numbers including fractions that terminate, such as 1/2 or 1/4, and all the integers, including all whole and natural numbers, it seems like a pretty easy bet that you should be able to draw any number of triangle pairs that satisfy these pretty simple rules. Shouldn't it?

And yet, for every possible combination you might care to test from the world of Euclidean geometry, there is only one isosceles triangle and one right triangle in existence that satisfy each rule (not counting simply scaling the triangles). The picture below reveals those two very similar, but different, triangles.

The only isosceles and right triangles with rational value side lengths that have identical perimeters and areas

That these are the only isosceles and right triangles that satisfy each of the rules listed above was demonstrated in an algebraic geometry proof put forward by Yoshinosuki Hirakawa and Hideki Matsumura in 2018 and published in the Journal of Number Theory. Here's the abstract from the paper:

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of p-adic abelian integrals.

Here's a much less technical description of how they cracked the problem:

The vexing question they grappled with was whether a pair of triangles with integral sides, a right-angle triangle and an isosceles triangle, can have the same perimeter and area.

No one had succeeded in proving the existence of such a pair.

The mathematicians converted the geometric question involving numerous triangles into an algebraic equation in their search for an answer.

They then applied diophantine geometry, a modern technique of mathematics, to the task to prove their theorem that “only one pair of such triangles exists.”

The new theorem, to be called the “Hirakawa-Matsumura theorem” from now on, states that there exists a unique pair of a rational right triangle and a rational isosceles triangle, which have the same perimeter and area. The unique pair consists of the right triangle with sides of lengths (377, 135, 352) and the isosceles triangle with sides of lengths (366, 366, 132), excluding pairs of similar triangles.

Hirakawa and Matsumura noted that in ancient Greece, mathematicians were preoccupied with geometric questions such as Pythagorean theorem.

“It seems likely that the theorem we proved was something the ancient Greeks also tried to figure out," Hirakawa said.

"In the end, after thousands of years, advanced modern mathematics has proved the theory," he added. "That is rare and what makes mathematics so exciting."

It's also part of what makes math fun, where as a bonus, we now know it would be a complete waste of time to seek out other isosceles-right triangle pairs to try to satisfy the original puzzle statement.

On the other hand, if you're looking for a different challenge that builds on right triangle math, you might enjoy exploring this Pythagorean-inspired problem!

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