1. **Problem statement:** Simplify the expression \(\frac{(x-a)(x-b)}{(c-a)(c-b)} + \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)}\). 2. **Formula and approach:** This expression is a sum of Lagrange basis polynomials for points \(a, b, c\). The sum of these basis polynomials is always 1 for any \(x\). 3. **Explanation:** Each term is constructed so that it equals 1 at one of the points \(a, b, c\) and 0 at the others. Summing all three gives a polynomial that is 1 everywhere. 4. **Verification:** Substitute \(x = a\), the first term becomes 1, others 0; similarly for \(x = b\) and \(x = c\). 5. **Conclusion:** Therefore, the entire expression simplifies to: $$ 1 $$