To solve the problem, we need to find the equation of a line that passes through the origin and intersects the lines \(x = 1\), \(y = 2\), and \(x + y = 4\) at points \(A\), \(B\), and \(C\) respectively, such that the product of the distances from the origin to these points is \(8\sqrt{2}\).
### Step-by-Step Solution:
1. **Define the Equation of the Line:**
Since the line passes through the origin, we can express it as:
\[
y = mx
\]
where \(m\) is the slope of the line.
2. **Find the Coordinates of Points A, B, and C:**
- **Point A (Intersection with \(x = 1\)):**
Substitute \(x = 1\) into the line equation:
\[
y = m \cdot 1 = m
\]
Thus, the coordinates of point \(A\) are \((1, m)\).
- **Point B (Intersection with \(y = 2\)):**
Substitute \(y = 2\) into the line equation:
\[
2 = mx \implies x = \frac{2}{m}
\]
Thus, the coordinates of point \(B\) are \(\left(\frac{2}{m}, 2\right)\).
- **Point C (Intersection with \(x + y = 4\)):**
Substitute \(y = mx\) into the equation \(x + y = 4\):
\[
x + mx = 4 \implies x(1 + m) = 4 \implies x = \frac{4}{1 + m}
\]
Then, substituting \(x\) back to find \(y\):
\[
y = m \cdot \frac{4}{1 + m} = \frac{4m}{1 + m}
\]
Thus, the coordinates of point \(C\) are \(\left(\frac{4}{1 + m}, \frac{4m}{1 + m}\right)\).
3. **Calculate the Distances OA, OB, and OC:**
- **Distance OA:**
\[
OA = \sqrt{(1 - 0)^2 + (m - 0)^2} = \sqrt{1 + m^2}
\]
- **Distance OB:**
\[
OB = \sqrt{\left(\frac{2}{m} - 0\right)^2 + (2 - 0)^2} = \sqrt{\frac{4}{m^2} + 4} = \sqrt{\frac{4 + 4m^2}{m^2}} = \frac{2\sqrt{1 + m^2}}{m}
\]
- **Distance OC:**
\[
OC = \sqrt{\left(\frac{4}{1 + m} - 0\right)^2 + \left(\frac{4m}{1 + m} - 0\right)^2}
\]
\[
= \sqrt{\frac{16}{(1 + m)^2} + \frac{16m^2}{(1 + m)^2}} = \sqrt{\frac{16(1 + m^2)}{(1 + m)^2}} = \frac{4\sqrt{1 + m^2}}{1 + m}
\]
4. **Set Up the Equation:**
According to the problem, we have:
\[
OA \cdot OB \cdot OC = 8\sqrt{2}
\]
Substituting the distances:
\[
\sqrt{1 + m^2} \cdot \frac{2\sqrt{1 + m^2}}{m} \cdot \frac{4\sqrt{1 + m^2}}{1 + m} = 8\sqrt{2}
\]
Simplifying:
\[
\frac{8(1 + m^2)^{3/2}}{m(1 + m)} = 8\sqrt{2}
\]
Dividing both sides by 8:
\[
\frac{(1 + m^2)^{3/2}}{m(1 + m)} = \sqrt{2}
\]
5. **Solve for m:**
Squaring both sides:
\[
\frac{(1 + m^2)^3}{m^2(1 + m)^2} = 2
\]
Cross-multiplying gives:
\[
(1 + m^2)^3 = 2m^2(1 + m)^2
\]
This is a polynomial equation in \(m\). By trial or numerical methods, we can find that \(m = 1\) satisfies the equation.
6. **Find the Equation of the Line:**
Substituting \(m = 1\) back into the line equation:
\[
y = mx \implies y = 1x \implies y = x
\]
### Final Answer:
The equation of the line is:
\[
\boxed{y = x}
\]
