Reducing an algebraic fraction, cyclic in three variables, to another

AI Thread Summary
The discussion revolves around reducing an algebraic fraction involving three variables in a cyclic manner. The original poster expresses difficulty in finding an elegant solution and shares their brute force attempt, which ultimately yields the correct answer. They provide a detailed algebraic manipulation that simplifies the expression to abc. The conversation seeks hints or methods for a more refined approach to achieve the same result. The focus remains on finding a more elegant solution to the problem presented.
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Homework Statement
If ##2s = a+b+c##, prove that $$\boxed{\pmb{\frac{1}{s-a}+\frac{1}{s-b}+\frac{1}{s-c}-\frac{1}{s}=\frac{abc}{s(s-a)(s-b)(s-c)}}}$$
Relevant Equations
I don't know if the following three formulae will be useful, all equivalent to one another and written out in different forms.
1. ##ab(a-b)+bc(b-c)+ca(c-a) = -(a-b)(b-c)(c-a)##
2. ##a^2(b-c)+b^2(c-a)+c^2(a-b)=-(a-b)(b-c)(c-a)##
3. ##a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)= (a-b)(b-c)(c-a)##
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Problem :
Let me copy and paste the problem statement as it appears in the text, as shown above.

Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything.

1639489036427.png


Ok so I have got the answer, with clumsy algebra and using brute force.Does someone have hints to an elegant approach?
 
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You can write the numerator as <br /> \begin{split}<br /> (s-a)(s-b)(s-(s-c)) + s(s-c)((s - a) + (s - b)) <br /> &amp;= c(s-a)(s-b) + s(s-c)(2s - a - b) \\<br /> &amp;= c(s-a)(s-b) + sc(s-c) \\<br /> &amp;= c(2s^2 - (a + b + c)s + ab) \\<br /> &amp;= c(2s^2 - 2s^2 + ab) \\<br /> &amp;= abc.<br /> \end{split}
 
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