# Triangles and Rectangles

Standard

Consider the following question by Bill Sands (asked in 1995):

Are there right triangles with integer sides and area, associated with rectangles having the same perimeter and area?

Try to test your intuition. The solution to this problem is NOT so simple. The solution was published by Richard K. Guy in 1995: http://www.jstor.org/stable/2974502

If you have a simpler solution, please write it in a comment below. Even I don’t understand much of the solution.

Life is never fair. So I made my life a mathematical fair.

### 3 responses

1. Gaurish, I will give it a try as well. It’s interesting. Let’s see if I can come up with a different solution!

Liked by 1 person

2. Suppose yes.
Up to scalar, the triple can be given by
$(m^2-n^2,2mn,m^2+m^2)$ for coprime
$m>n\ge 1$,
and the rectangle needs rational sides satisfying
$x+y=m(m+n)$ and
$xy=mn(m-n)(m+n)$.

Since m>1, m has a prime factor p. Assume
$p^i||x$,
$p^j||y$.
Then
$p^{\text{min}(i,j)}||m$ from the sum,
$p^{i+j}||m$ from the product, and
$i+j=\text{min}(i,j)>0$.
This is impossible.

I see that the paper considers integer triangles in general. They begin with a similar formula by Brahmagupta. My trick fails here; the parameters a and b need not be coprime. It makes sense that Guy and Bremmer use more powerful tools from algebraic geometry to study the problem.

Liked by 1 person

• So the trick was to use the Pythagorean triplets for the integer right triangle. Nice 🙂
But, shouldn’t there be another case when i=j=0?

Like