We are given the equation:

$a^2 + b^2 + c^2 - 2(a + b + c) + 3 = 0$

We need to find the value of $a \cdot b \cdot c$.

Step 1: Simplify the equation

We can rearrange the equation as follows:

$a^2 + b^2 + c^2 - 2a - 2b - 2c + 3 = 0$

Now, let's try to complete the square for the quadratic terms involving $a$, $b$, and $c$.

Step 2: Completing the square

We'll group the terms involving $a$, $b$, and $c$:

$(a^2 - 2a) + (b^2 - 2b) + (c^2 - 2c) + 3 = 0$

Complete the square for each term:

$(a - 1)^2 - 1 + (b - 1)^2 - 1 + (c - 1)^2 - 1 + 3 = 0$

Simplify the constants:

$(a - 1)^2 + (b - 1)^2 + (c - 1)^2 = 0$

Step 3: Solve for $a$, $b$, and $c$

Since squares of real numbers are non-negative, the only solution to this equation is:

$(a - 1)^2 = 0, \quad (b - 1)^2 = 0, \quad (c - 1)^2 = 0$

This implies:

$a = 1, \quad b = 1, \quad c = 1$

Step 4: Find $a \cdot b \cdot c$

Now that we know $a = 1$, $b = 1$, and $c = 1$, we can find the product:

$a \cdot b \cdot c = 1 \cdot 1 \cdot 1 = 1$

Thus, the value of $a \cdot b \cdot c$ is $\boxed{1}$.

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Would you like more details on any part of this solution?

Here are 5 related questions:

1. What if one of the values for $a$, $b$, or $c$ was negative? How would that change the equation?
2. How do you complete the square for quadratic equations in general?
3. Can you explain how to solve other types of quadratic systems involving three variables?
4. How would the problem change if the constants in the equation were different?
5. What happens if you add an additional term like $abc$ in the original equation?

Tip: Completing the square is a powerful technique not only for solving equations but also for analyzing quadratic expressions geometrically!