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. **Understand the given information:** We know both $a - b$ and $b - c$ equal 2. This means:\n$$a - b = 2 \quad \text{and} \quad b - c = 2.$$\n\n3. **Find $a - c$:** Using the properties of subtraction,\n$$a - c = (a - b) + (b - c) = 2 + 2 = 4.$$\n\n4. **Substitute values into the expression:**\n$$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{2^2 + 2^2}{4^2} = \frac{4 + 4}{16} = \frac{8}{16}.$$\n\n5. **Simplify the fraction:**\n$$\frac{8}{16} = \frac{1}{2}.$$\n\n**Final answer:** $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2} = \frac{1}{2}.$$