A New Kind of Science: The NKS Forum > Pure NKS > Pythagorean Triangles Arranged in Order
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New Zealand

Registered: Aug 2010
Posts: 1

Pythagorean Triangles Arranged in Order

I haven¡¦t seen this mentioned on any maths-related websites and believe a lot of people would find it interesting.

Most interesting is the third tab of the Excel file (page 3 of the pdf) which extends the original rows and columns upwards and to the left and right.

This gives four sets of triangles, either with three positive sides or with one or both of the legs negative.

To preserve the formulae, the diagonal axes are triangles with one side of zero length.

The second tab of the Excel file (page 2 of the pdf) shows the in-circle radii of the triangles on the first tab.

I was interested to note that summing the rows on the second tab gives the sum of the first n squares.

Triangle notes
Methods of Generating Pythagorean Triangles
Create a column of Pythagorean triangles for the nth odd number, X:
„h
A = n x 4
„h
B = A + X^2
„h
C = X^2 + 2X
The second triangle follows:
„h
D = A + ((n + 1) x 4)
„h
E = D + X^2
„h
F = C + 2X
And the third:
„h
G = D + ((n + 2) x 4)
„h
H = E + X^2
„h
I = F + 2X
And so on; see Excel spreadsheet ¡§tri gen¡¨.
Patterns in Pythagorean Triples
Hypotenuse and Longest side are consecutive
This sequence is Column B of the spreadsheet, where n = 1
The two legs are consecutive -
These triangles appear on a line bisecting the top angle of the triangular spreadsheet; highlighted yellow on the spreadsheet.
More Patterns
Primitive triangles whose longest side and hypotenuse differ by 2; these triangles appear in a line as well ¡V they are the first triangle of each column in the spreadsheet.
The inradii are shown on the second tab. For a column generated for the nth odd number X the inradii are the multiples of X in numerical order.

Attachment: tri gen 23-08.pdf