Proof:

Let's assume that the complex numbers z1, z2, and z3 are represented as follows:

z1 = x1 + y1i,
z2 = x2 + y2i,
z3 = x3 + y3i,

where x1, y1, x2, y2, x3, and y3 are real numbers.

We are given that the expression (z3 - z1) / (z2 - z1) is a real number.

Step 1: Simplify the expression (z3 - z1) / (z2 - z1)

(z3 - z1) / (z2 - z1) = [(x3 + y3i) - (x1 + y1i)] / [(x2 + y2i) - (x1 + y1i)]
= [(x3 - x1) + (y3 - y1)i] / [(x2 - x1) + (y2 - y1)i]

To make the expression a real number, the imaginary part must be zero. Therefore, we have:

y3 - y1 = 0,
y3 = y1.

Step 2: Rewrite the complex numbers z1, z2, and z3

Using the result from Step 1, we can rewrite z3 as:

z3 = x3 + y1i.

Now, let's rewrite the expression (z3 - z1) / (z2 - z1) using the new representation of z3:

(z3 - z1) / (z2 - z1) = [(x3 + y1i) - (x1 + y1i)] / [(x2 + y2i) - (x1 + y1i)]
= [(x3 - x1) + (y1 - y1)i] / [(x2 - x1) + (y2 - y1)i]
= (x3 - x1) / (x2 - x1).

Step 3: Interpretation

The expression (x3 - x1) / (x2 - x1) is a real number, which means the imaginary part of this expression is zero. Therefore, we have:

0 = Im[(x3 - x1) / (x2 - x1)]
= (x3 - x1)(x2 - x1)^(-1) - (x3 - x1)(x2 - x1)^(-1)
= 0.

This implies that the imaginary part of (z3 - z1) / (z2 - z1) is zero, which means the expression represents a real number.

Conclusion:

Since the expression (z3 - z1) / (z2 - z1) is a real number, we have shown that the points represented by the complex numbers z1, z2, and z3 are collinear.