To solve the problem, we need to find the length of the side of an equilateral triangle ABC, where point B lies on one parallel line, point C lies on the other parallel line, and point A is located between the two lines at a distance 'd' from one of the lines. The distance between the two parallel lines is given as 1 unit.
### Step-by-Step Solution:
1. **Understanding the Setup**:
- Let the two parallel lines be represented as \( y = 0 \) and \( y = 1 \).
- Point A is located at a distance \( d \) from the line \( y = 0 \). Thus, the coordinates of point A can be expressed as \( A(0, d) \).
- Since the distance between the two lines is 1 unit, the distance from point A to the line \( y = 1 \) is \( 1 - d \).
2. **Positioning Points B and C**:
- Let point B be on the line \( y = 0 \) and point C be on the line \( y = 1 \).
- We can denote the coordinates of point B as \( B(x_B, 0) \) and point C as \( C(x_C, 1) \).
3. **Using the Properties of an Equilateral Triangle**:
- In an equilateral triangle, all sides are equal. Therefore, the lengths \( AB \), \( BC \), and \( CA \) must all be equal.
- The length of side \( AB \) can be calculated using the distance formula:
\[
AB = \sqrt{(x_B - 0)^2 + (0 - d)^2} = \sqrt{x_B^2 + d^2}
\]
- The length of side \( AC \) is:
\[
AC = \sqrt{(x_C - 0)^2 + (1 - d)^2} = \sqrt{x_C^2 + (1 - d)^2}
\]
- The length of side \( BC \) is:
\[
BC = \sqrt{(x_C - x_B)^2 + (1 - 0)^2} = \sqrt{(x_C - x_B)^2 + 1}
\]
4. **Setting Up the Equations**:
- Since \( AB = AC \), we can set up the equation:
\[
\sqrt{x_B^2 + d^2} = \sqrt{x_C^2 + (1 - d)^2}
\]
- Squaring both sides gives:
\[
x_B^2 + d^2 = x_C^2 + (1 - d)^2
\]
- This simplifies to:
\[
x_B^2 - x_C^2 = (1 - d)^2 - d^2
\]
- The right-hand side simplifies to:
\[
(1 - 2d + d^2) - d^2 = 1 - 2d
\]
- Thus, we have:
\[
x_B^2 - x_C^2 = 1 - 2d
\]
5. **Using the Equivalence of AB and BC**:
- Now, since \( AB = BC \), we set up another equation:
\[
\sqrt{x_B^2 + d^2} = \sqrt{(x_C - x_B)^2 + 1}
\]
- Squaring both sides gives:
\[
x_B^2 + d^2 = (x_C - x_B)^2 + 1
\]
- Expanding the right-hand side:
\[
x_B^2 + d^2 = x_C^2 - 2x_B x_C + x_B^2 + 1
\]
- This simplifies to:
\[
d^2 = x_C^2 - 2x_B x_C + 1
\]
6. **Solving the Equations**:
- We now have two equations:
1. \( x_B^2 - x_C^2 = 1 - 2d \)
2. \( d^2 = x_C^2 - 2x_B x_C + 1 \)
- By substituting \( x_C^2 \) from the first equation into the second, we can solve for \( s \) (the side length of the triangle).
7. **Final Calculation**:
- After manipulating the equations, we find that:
\[
s^2 = \frac{4}{3}(d^2 - d + 1)
\]
- Therefore, the length of the side \( s \) of the equilateral triangle is:
\[
s = \frac{2}{\sqrt{3}} \sqrt{d^2 - d + 1}
\]
### Conclusion:
The length of the side of the equilateral triangle ABC is given by:
\[
s = \frac{2}{\sqrt{3}} \sqrt{d^2 - d + 1}
\]
