There seems to be nothing special about the given expression since the cyclic symmetry is not present. So it is not an identity with constant value for all a,b,c.
So, we can have variable value of the expression depending on the values of a,b and c. For example, for a=0,b=1 and c=2 the sum is -1.5 and that for a=1, b=4 and c=2 is 1.167.
Restoring cyclic symmetry in the above expression will give us the following identity :
a/(a−c)(a−b)+b/(b−c)(b−a)+c/(c−a)(c−b) = 0
This can be proved by simple addition of the LHS.
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