Triangles and Rectangles

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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.

3 responses

  1. 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.

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