1. **State the problem:** We are given that $a - b = b - c = 2$ and need to find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}.$$ 2. **Use the given equalities:** Since $a - b = 2$ and $b - c = 2$, substitute these values into the numerator: $$ (a - b)^2 + (b - c)^2 = 2^2 + 2^2 = 4 + 4 = 8.$$ 3. **Find $a - c$:** Using the relation, $$a - c = (a - b) + (b - c) = 2 + 2 = 4.$$ 4. **Calculate the denominator:** $$ (a - c)^2 = 4^2 = 16.$$ 5. **Form the fraction and simplify:** $$ \frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{8}{16} = \frac{\cancel{8}}{\cancel{16}} = \frac{1}{2}.$$ 6. **Final answer:** $$\boxed{\frac{1}{2}}.$$