Problem 1253

1. Identify the coordinates of the opposite vertices of the square:
The given points are $(1, 2)$ and $(7, 4)$.

2. Calculate the length of the diagonal of the square:
The length of the diagonal can be found using the distance formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
Substituting the given points $(1, 2)$ and $(7, 4)$:
$$d = \sqrt{(7 - 1)^2 + (4 - 2)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40}$$

3. Relate the diagonal to the side length of the square:
For a square with side length $s$, the diagonal $d$ is given by:
$$d = s\sqrt{2}$$
Given that the diagonal $d = \sqrt{40}$, we can set up the equation:
$$\sqrt{40} = s\sqrt{2}$$

4. **Solve for the side length $s$:**
$$s = \frac{\sqrt{40}}{\sqrt{2}} = \sqrt{\frac{40}{2}} = \sqrt{20}$$

5. Calculate the area of the square:
The area $A$ of a square with side length $s$ is given by:
$$A = s^2$$
Substituting $s = \sqrt{20}$:
$$A = (\sqrt{20})^2 = 20$$

Conclusion:
$$\boxed{20}$$