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        riddles >> medium >> semi perimeter a square
        
(Message started by: Christine on Mar 30^th, 2014, 1:26pm)

Title: semi perimeter a square
Post by Christine on Mar 30^th, 2014, 1:26pm

Can you find a triangle whose area, sides are integers and the semi perimeter is a square?

Is there an algorithm to generate them?

Title: Re: semi perimeter a square
Post by pex on Mar 30^th, 2014, 5:27pm

on 03/30/14 at 13:26:51, Christine wrote:

Can you find a triangle whose area, sides are integers and the semi perimeter is a square?

Yes. The simplest triangle with integer sides and area that I can think of has sides 3,4,5. Its semiperimeter is 6, so scaling the whole thing up by a factor of 6 works: the 18,24,30 triangle fits the requirements.

Title: Re: semi perimeter a square
Post by Christine on Apr 10^th, 2014, 3:18pm

on 03/30/14 at 17:27:03, pex wrote:

Thanks. I'm looking for a primitive Heronian triangle

Title: Re: semi perimeter a square
Post by towr on Apr 10^th, 2014, 11:13pm

The smallest primitive with those properties is 8,5,5 (which is just 2 (5,4,3)'s put together)

You can generate a list based on http://en.wikipedia.org/wiki/Heronian_triangle#Exact_formula_for_Heronian_triangles

e.g.
Python
Code:

from fractions import gcd

for m in range(1000):
for n in range(1,m+1):
for k in range(int(((m**2*n)/(2*m+n))**0.5-1), int((m*n)**0.5)+1):
if gcd(m,gcd(n,k))==1 and m*n > k**2 >= (m**2*n)/(2*m+n) and m>=n>=1:
a = n*(m**2+k**2)
b = m*(n**2+k**2)
c = (n+m)*(m*n-k**2)
g = gcd(a,gcd(b,c))
a /= g
b /= g
c /= g
s = (a+b+c)/2
if int(s**0.5)**2==s:
print (m,n,k, a,b,c, (a+b+c)/2)

(697, 657, 104) is the first I've found to also have a square area.