Geometry construction exercise

Question

I have another construction Problem.

What is given is a point A and two streight lines f and g. One should construct a Rhombus ABCD with side lenght 4cm. B should be on f and his midpoint on g.

See Picture: 1

So far I drawed a circle of ray 4cm centered in A. So one of the two intersections with f has to be B. The second cirrcle is also of 4 cm starting from one of the two intersection points (since all the sides have to be the same lenght. But this attempt is not really walid since you cant be sure wich of the two intersections is bin the first place).

Can someone see how to get the Rhombus from here?

Answer 1

Draw the red circle! and mark $B$ on $f$. Draw the green circle and mark $M$.
$AM$ and $BM$ are the half diagonals of your rhombus.
Of course point $N$ can also be another solution.

Answer 2

Let me propose the solution initially following your sketch.