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A Most Curious Algebraic Identity

I recently found a very interesting Algebraic Identity: \[xyz+(x+y)(y+z)(z+x)=(x+y+z)(xy+yz+zx)\]

What's so special about it? Note that going from one side of the equality to the other, all products are switched with sums, and all sums are switched with products!

This may be seen a bit easier if I rewrite it as follows: \[ \begin{align*} &\color{white}{)}x\color{red}{\times} y\color{red}{\times }z \color{white}{)}\color{blue}{+} \color{grey}{(}x\color{blue}{+}y\color{grey}{)}\color{red}{\times}\color{grey}{(}y\color{blue}{+}z\color{grey}{)}\color{red}{\times} \color{grey}{(}z\color{blue}{+}x\color{grey}{)}\\ =&\color{grey}{(}x\color{blue}{+}y\color{blue}{+}z\color{grey}{)}\color{red}{\times}\color{grey}{(}x\color{red}{\times} y\hspace{0.9ex}\color{blue}{+}\hspace{0.9ex}y\color{red}{\times }z\color{white}{)}\color{blue}{+}\hspace{0.9ex}z\color{red}{\times }x\color{grey}{)} \end{align*}\]

Cool!

Note by Daniel Liu
2 years, 11 months ago

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Yes; I really like this identity too. For example it can be used to prove that \(r_1 + r_2 + r_3 - r = 4R\) (from Incircles and Excircles). If we substitute \(x = s-a\) etc., then

\[s(s-b)(s-c)+s(s-c)(s-a)+s(s-a)(s-b)-(s-a)(s-b)(s-c) = abc\]

where \(s\) is the semi-perimeter, and this reduces nicely using area formulas to the desired relationship.

Michael Ng - 2 years, 11 months ago

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@Michael Ng created this problem which uses the identity.

Calvin Lin Staff - 2 years, 10 months ago

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thanks @Daniel Liu i used this to solve problems like this i wrote a solution using this identity, and i'm thinking about a problem with this identity, will post soon!

Aareyan Manzoor - 2 years, 10 months ago

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F B U L O U S !!!!!!

Atanu Ghosh - 2 years, 2 months ago

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I didn't realise that- thanks. It should inspire some good problems :)

Curtis Clement - 2 years, 11 months ago

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Most awesome discoveries ever!Thanks,this must help a lot.

Frankie Fook - 2 years, 9 months ago

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Thanks. I too did not realize this very useful identity.

Niranjan Khanderia - 2 years, 10 months ago

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I didn't examine. Good for making questions. However, it could have been found by people in the past.

Lu Chee Ket - 2 years, 10 months ago

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Yea, I'm just saying that I just noticed it. I most likely was not the person who discovered it (as seen by the comment by Michael Ng)

Daniel Liu - 2 years, 10 months ago

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I had just expanded both sides to compare. Should be correct. Do not feel disappointed by what I guessed. You could be the first person to find this. Congratulation!

Lu Chee Ket - 2 years, 10 months ago

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It's following the rules of principle of duality

Akhil Bansal - 2 years, 2 months ago

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No, that is not the principle of duality.

Calvin Lin Staff - 2 years, 2 months ago

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I mean l'll bit similar to that

Akhil Bansal - 2 years, 2 months ago

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