40. Divide the straight line $A B$ at point $H$ into two segments such that the larger segment is the mean proportional between the whole line and its smaller segment.
40. Proposition XI of the second book states: to divide the straight line $A B$ at point $H$ into two parts such that the rectangle contained by the whole straight line $A B$ and one of its parts is equal to the square constructed on the other part.

Euclid's Construction. Construct the square $A B C D$ on the straight line $A B$, bisect $A C$ at point $E$, and connect $B$ with $E$. Extend $A C$ to point $F$ so that $B E = F E$. Construct the square $A F G H$ on $A F$ and extend $G H$ to meet $D C$ at point $K$. The straight line $A B$ will be divided such that

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-080.jpg?height=488&width=325&top_left_y=384&top_left_x=1391)

Fig. 16.

Indeed: Let $H A = x$ and $A B = a$, then $B H = a - x$; since $A E = \frac{a}{2}$, then $F E = x + \frac{a}{2} = B E$, but $B E^2 = A B^2 + A E^2$, or

$$
x^2 + \frac{a^2}{4} + a x = a^2 + \frac{a^2}{4}
$$

from which

$$
x^2 = a^2 - a x = a(a - x)
$$

Euclid's proof is given from different considerations.