geometric mean - Höhensatz
$${ h^2 = p \cdot q}$$ , where p and q are the segments formed by the height h from point C onto the opposite side c.
Rearranging, it can be written
$${ \frac{h}{q} = \frac{p}{h}}$$
where h is called the geometric mean. $${ h = \sqrt{p \cdot q}}$$ In this case, the geometric mean is the height of the right-angled triangle over the side c.
constructing roots using the “Höhensatz”
The root of 12 can be construted by using the factors of 12, say 3 and 4, by drawing the line segments with length 3 and 4 side by side. Then, after finding the midpoint of the two segments at 3.5, drawing the semi-circle with radius 3.5 and diameter 7 (3 + 4) and drawing the perpendicular height where the two sements meet. The height, according to the “Höhensatz”, has the length $\sqrt{12}$.