Contruction exercise

Question

I am struggling with this exercise.

Given is the Midpoint M of a square ABCD and two straight lines g and h. One should construct the square ABCD having A on g and C on h 1

My attempts: sketch

I know I am looking for A and C to have the same distance to M.

Someone knows how to do this? Bests

Answer 1

Draw a line from $M$, parallel to one of those straight lines, let's say parallel to $h$ and call the point $P$ the intersection with the line $g$ .
Mark $A$ on $g$ such that $AP=PO$.

Line $AMC$ is the diagonal of your square.

Answer 2

Let $g\cap h=\{K\}$ and $L$ be placed on the ray $KM$ such that $M$ is a mid-point of $KL$.

Now, let $h'||h$ and $L\in h'$.

Also, let $g'||g$ and $L\in g'$.

Let $h'\cap g=\{A\}$ and $g'\cap h=\{C\}$.

Thus, $ALCK$ is a parallelogram, which gives that $M$ is a mid-point of $AC$.

Now, easy to construct a square $ABCD$.