To solve the equation \((x - 2)^3 = (x - 2)^{12}\), we can start by simplifying the terms.

First, we recognize that this equation can be simplified by dividing both sides by \((x - 2)^3\), assuming \(x - 2 \neq 0\):

\[
1 = (x - 2)^{12 - 3}
\]
This simplifies to:

\[
1 = (x - 2)^9
\]

Next, we solve for \(x - 2\):

Taking the 9th root of both sides gives us:

\[
x - 2 = 1 \quad \text{or} \quad x - 2 = -1
\]

This leads us to two cases:

1. \(x - 2 = 1\)
\[
x = 3
\]

2. \(x - 2 = -1\)
\[
x = 1
\]

Now, we must also consider the case where \((x - 2)^3 = 0\). This occurs when:

\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]

Now, let's summarize all the solutions:

The solutions to the equation \((x - 2)^3 = (x - 2)^{12}\) are:

\[
x = 1, \quad x = 2, \quad x = 3
\]

These are the values of \(x\) that satisfy the original equation.