1. **State the problem:** Simplify the expression $$\frac{1}{1+x+y-1} + \frac{1}{1+y+z-1} + \frac{1}{1+z+x-1}$$ given that $$xyz=1$$. 2. **Simplify each denominator:** $$1+x+y-1 = x+y$$ $$1+y+z-1 = y+z$$ $$1+z+x-1 = z+x$$ So the expression becomes: $$\frac{1}{x+y} + \frac{1}{y+z} + \frac{1}{z+x}$$ 3. **Use the condition $$xyz=1$$:** This means $$z=\frac{1}{xy}$$. 4. **Rewrite the denominators using $$z=\frac{1}{xy}$$:** $$y+z = y + \frac{1}{xy} = \frac{y^2x + 1}{xy}$$ $$z+x = \frac{1}{xy} + x = \frac{1 + x^2 y}{xy}$$ 5. **Rewrite the expression with common denominators:** $$\frac{1}{x+y} + \frac{1}{\frac{y^2 x + 1}{xy}} + \frac{1}{\frac{1 + x^2 y}{xy}} = \frac{1}{x+y} + \frac{xy}{y^2 x + 1} + \frac{xy}{1 + x^2 y}$$ 6. **No further simplification is straightforward without additional constraints, so the simplified form is:** $$\boxed{\frac{1}{x+y} + \frac{xy}{y^2 x + 1} + \frac{xy}{1 + x^2 y}}$$