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On generating integer sided triangles with an integer area perimeter ratio

On generating integer sided triangles with an integer area perimeter ratio

This blog post contains come comments about finding all triangles whose sizes are integers and the area/perimeter ratio is the integer r. This is related to Project Euler problem 283.

The blog post contains a nice analysis and some pseudocode for generating all possible (a, b, c) size triplets for the given ratio r. Unfortunately both the analysis and the pseudocode contains some mistakes.

Here is the corrected version:

def find_triangles_correct(r):
 for u in divisors of 2r:
 for v in 1 to floor(sqrt(3) * u):
 if gcd(u, v) == 1:
 lhs = 4*r*r*(u*u+v*v)
 for d1 in positive divisors of lhs:
 if d1 <= lhs / d1:
 d2 = lhs / d1
 if not (u < v and d1*u < (2r(v*v-u*u))):
 if ((d1+2ru) mod v) = 0 and ((d2+2ru) mod v) = 0:
 a = (d1 + 2ru) / v + 2rv / u
 b = (d2 + 2ru) / v + 2rv / u
 c = (d1 + d2 + 4ru) / v
 yield a, b, c

Some mistakes and other inaccuracies in the original:

• The variable r were silently renamed to m.
• The formulas for a, b and c were not included.
• The analysis incorrectly states that v must be smaller than floor(sqrt(3)*u)). The correct statement is: v must be smaller than sqrt(3)*u. Before the correction some solutions such as (6, 8, 10) for r=1 were not generated.
• The condition d1 <= lhs / d1 had a >= instead.

Please note that the algorithm yields each triangle once, in an unspecified order of a, b and c within the triangle.

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