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| Even Perimeter (Posted on 2009-01-16) |
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Prove that if a triangle's area and sides are all integers, its perimeter must be even.
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Submitted by Praneeth
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Solution:
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Credit to Daniel for the solution:
Let A be the area
a,b,c be the sides
p be the perimeter
s=p/2
then
A^2=s(s-a)(s-b)(s-c)
A^2=p(p-2a)(p-2b)(p-2c)/16
16*A^2=p(p-2a)(p-2b)(p-2c)
now if p is odd then p=2k+1 for integer k
and
16*A^2=(2k+1)(2(k-a)+1)(2(k-b)+1)(2(k-c)+1)
left side is even but right side has all odd factors and thus
can not be even so therefore p can not be odd and thus must be even.
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