1. **Stating the problem:** We are given that $a - b = b - c = 2$ and need to find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}.$$\n\n2. **Understanding the given information:** Since $a - b = 2$ and $b - c = 2$, both differences are equal to 2.\n\n3. **Calculate the numerator:**\n$$(a - b)^2 + (b - c)^2 = 2^2 + 2^2 = 4 + 4 = 8.$$\n\n4. **Calculate the denominator:**\nFirst, find $a - c$. Using the given equalities,\n$$a - c = (a - b) + (b - c) = 2 + 2 = 4.$$\nThen,\n$$(a - c)^2 = 4^2 = 16.$$\n\n5. **Compute the fraction:**\n$$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{8}{16} = \frac{1}{2}.$$\n\n**Final answer:** $$\boxed{\frac{1}{2}}.$$