I have found the following geometric construction exercise:
Given a point $P$, a line $\ell$, a length $m$ and a radius $r$, draw the circumference with radius $r$ which goes through point P and intersect line $\ell$ in such a way that it creates a chord of length $m$.
Now, I think one should first draw the circle of radius $r$ and center $P$: then clearly the center of the circle we want will be one of the points of this circle.
My problem is now to find this point without using neither trigonometry nor analytic geometry (which I already know how to do): of the points of the circle $(P,r)$ how do I find out which ones also cut on the line a chord of the desired length? Thanks.
Make a drawing "at the side" to figure out what the distance $d$ between the center of the chord and the origin of the circle will be. In order to achieve this, draw a triangle with the side lengths $r$, $r$ and $m,$ and let $d$ be the distance between the vertex where the sides of length $r$ meet and the center of the side of length $m.$
Now draw a circle with radius $r$ and origin $P$. The origin of the circle you are looking for will lie on this circle.
Draw lines parallel to $\ell$ with distance $d$ to $\ell$ on both sides of $\ell.$ One of those two lines must intersect the circle you have just drawn, otherwise there is no solution. The origin of the circle you are looking for will also lie on one of those lines, because it must have precisely the distance $d$ to $\ell$ to ensure the correct chord length.
So the intersection of the circle with radius $r$ and origin $P$ and the line parallel to $\ell$ having a distance of $d$ to $\ell$ is the origin of the circle you are looking for.
There may be several solutions.