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Since the Pythagorean Theorem applies to right triangles, it can be stated:

In triangle $ABC$, the length of whose sides are $a$, $b$ and $c$, $a^2+b^2=c^2$ if and only if $\cos(\angle C)=0$.

I recently proved the following related result:

In triangle $ABC$, the length of whose sides are $a$, $b$ and $c$, $a(a+k)+b(b+k)=c(c+k)$ if and only if $\cos(\angle C)=−k/(a+b+c+k)$.

Is this a known result and, if so, where has it appeared?

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It looks to me like it could probably be deduced from the law of cosines. –  Kevin Carlson Oct 27 '12 at 3:18
    
What exactly is $k$? –  EuYu Oct 27 '12 at 3:18
2  
Wasn't this answered in a comment on MathOverflow? Yes, it's here: mathoverflow.net/questions/110712/… –  Gerry Myerson Oct 27 '12 at 3:23
    
If you cross-post, please include a link to the other post so that effort isn't unnecessarily duplicated. –  joriki Oct 27 '12 at 6:18

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