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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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    If you cross-post, please include a link to the other post so that effort isn't unnecessarily duplicated.2012-10-27

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