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 $$a - b = 2 \quad \text{and} \quad b - c = 2.$$\n\n3. **Calculate the numerator:** $$ (a - b)^2 + (b - c)^2 = 2^2 + 2^2 = 4 + 4 = 8.$$\n\n4. **Find $a - c$:** Since $a - c = (a - b) + (b - c)$, we have $$a - c = 2 + 2 = 4.$$\n\n5. **Calculate the denominator:** $$ (a - c)^2 = 4^2 = 16.$$\n\n6. **Form the fraction and simplify:** $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{8}{16} = \frac{\cancel{8}}{\cancel{16}} = \frac{1}{2}.$$\n\n**Final answer:** $$\boxed{\frac{1}{2}}.$$