The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
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1. **Identify the given information and the goal:**
   - We are given a square \(ABCD\) with points \(A\), \(B\), and \(C\) lying on three parallel lines.
   - The distances between these points are indicated in the figure.
   - We need to find the area of the square \(ABCD\).

2. **Understand the geometric relationships:**
   - Since \(ABCD\) is a square, all sides are equal, and all angles are \(90^\circ\).
   - The distances between the points on the parallel lines are given as follows:
     - \(AF_A = 5\)
     - \(BF_C = 5\)
     - \(BF_A = 7\)
     - \(F_CC = 7\)

3. **Use the Pythagorean Theorem to find the side length of the square:**
   - Consider the right triangle \(BF_CC\):
     - The legs of the triangle are \(BF_C = 5\) and \(F_CC = 7\).
     - The hypotenuse \(BC\) can be found using the Pythagorean Theorem:
     \[
     BC^2 = BF_C^2 + F_CC^2 = 5^2 + 7^2 = 25 + 49 = 74
     \]
     - Therefore, the length of \(BC\) is:
     \[
     BC = \sqrt{74}
     \]

4. **Calculate the area of the square:**
   - The area of a square is given by the square of the side length.
   - Since \(BC\) is a side of the square \(ABCD\), the area \(A\) of the square is:
     \[
     A = BC^2 = 74
     \]

The final answer is \(\boxed{74}\).