Quote:
Originally Posted by nemo
A cool fact I proved accidentally while taking a number theory class is that the radius of the circle inscribed by any Pathagorean triangle is always an integer.
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I missed this on the first read.
Generating pythogorean triples in the usual way by setting m and n to relatively-prime naturals...
a=m^2-n^2, b=2mn, c=m^2+n^2,
and calculating the in-radius in the usual way ...
s=(a+b+c)/2=m^2+mn=m(m+n)
r^2=(s-a)(s-b)(s-c)/s=(n^2+mn)(m^2-mn)(mn-n^2)/[m(m+n)]=
(m^2n^4-m^4n^2-mn^5+m^3n^3)/[m(m+n)]=
[m^2n^2(n^2-m^2)-mn(n^2-m^2)]/[m(m+n)]=
[(m^2n^2-mn)(n^2-m^2)]/[m(m+n)]=
(mn^2-n)(n-m) which is an integer.
My way is inelegant as usual. It's been a while since I was a math major, but I think that could be made rigorous.
I'm sure yours was far more elegant, considering you did it by "accident."