The complexity of algorithms of constructions by compass and straightedge

The article deals with the following problem. Assume that there are two points $A$ and $B$ on the plane, and a natural number $n$ is given. Our aim is to find the third point $C$ on the line containing $A$ and $B$ so that the length $AC$ is $n$ times larger than the length $AB$ using only a compass and a straightedge. During every step we can either construct a straight line containing two constructed points, or a circle with a constructed point as a center and with a radius equal to the distance between two constructed points. Intersections of constructed lines and circles form new constructed points. Denote the minimal number of steps necessary to solve this problem using only the compass as $\textup{C}(n)$, and the minimal number of steps necessary to solve this problem using both the compass and the straightedge as $\textup{CS}(n)$. We want to estimate the asymptotic behavior of the functions $\textup{C}(n)$ and $\textup{CS}(n)$. Our main result is the following: there exist constants $c_1, c_2>0$ such that a) $c_1\ln n\le\textup{C}(n)\le c_2 \ln n$, b) $c_1\ln\ln n\le\textup{CS}(n)\le\frac{c_2\ln n}{\ln\ln n}$. The most interesting result is obtained in connection with the lower bound of $\textup{CS}(n)$, where purely algebraic notions, such as the height of a number etc., arise quite unexpectedly.