To square a given triangle, we must construct a square whose area equals the area of the given triangle. First, identify the base ($b$) and height ($h$) of the triangle. The area of the triangle is given by the formula:

$$\frac{1}{2}bh$$

We need a square with the same area, $\frac{1}{2}bh$. The side of this square must be:

$$\sqrt{\frac{bh}{2}}$$

To achieve this, we can first construct a rectangle with sides $b$ and $\frac{h}{2}$. The area of this rectangle will be:

$$b \times \frac{h}{2} = \frac{bh}{2}$$

which exactly matches the triangle's area.

Next, we must convert this rectangle into a square of equal area. The side of such a square is the geometric mean of the rectangle's sides, $b$ and $\frac{h}{2}$. We can find this geometric mean using the semicircle method.

Draw a semicircle with a diameter equal to the sum of $b$ and $\frac{h}{2}$. Mark a point on the diameter that divides it into two segments of lengths $b$ and $\frac{h}{2}$. Draw a perpendicular from this point to the circumference of the semicircle.

The length of this perpendicular is:

$$\sqrt{b \times \frac{h}{2}}$$

which is the required side length. Finally, construct a square using this length. This square will have an area equal to the given triangle.