1. **State the problem:** Given that $a - b = b - c = 2$, find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}.$$\n\n2. **Use the given information:** We know $a - b = 2$ and $b - c = 2$.\n\n3. **Calculate the numerator:** Substitute the values into the numerator:\n$$ (a - b)^2 + (b - c)^2 = 2^2 + 2^2 = 4 + 4 = 8. $$\n\n4. **Calculate the denominator:** First find $a - c$. Since $a - b = 2$ and $b - c = 2$, then\n$$ a - c = (a - b) + (b - c) = 2 + 2 = 4. $$\nNow square it:\n$$ (a - c)^2 = 4^2 = 16. $$\n\n5. **Form the fraction and simplify:**\n$$ \frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{8}{16} = \frac{\cancel{8}}{\cancel{16}} = \frac{1}{2}. $$\n\n**Final answer:** $$\boxed{\frac{1}{2}}.$$