Quadruple of Pythagorean triples with same area

Solution 1:

It says here, "One can also find quartets of right triangles with the same area. The quartet having the smallest known area is $$(111, 6160, 6161), (231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458)$$ with area $341880$ (Beiler 1966, p. 127). Guy (1994) gives additional information."

The references are Beiler, A. H., "The Eternal Triangle," Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966, and Guy, R. K., "Triangles with Integer Sides, Medians, and Area." §D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994. But there is a 3rd edition of Guy's book.

I note that these triangles are not all primitive, but you didn't ask for that.

Solution 2:

Look at A055193 i oeis.org. I added a little in EXAMPLE about the area that gives five triangles with the same area Sture Sjöstedt sture.sjostedt(at)spray.se