An Euler's identity

Euler has the following identity in one of his works:
(a+b)/c*(b+c)/a*(c+a)/b = 2+(a+b)/c+(b+c)/a+(c+a)/b for all a,b,c not equal to 0. So I wonder if any one happens to know whether this identity has a general case for n variables or at least like 4 variables (a, b, c, d).