Constructing triangle $\triangle ABC$ given median to the side $c$ and angles $\alpha$ and $\beta$

Question

Constructing triangle $\triangle ABC$ given median to the side $c$ and angles $\alpha$ and $\beta$

I started with the median. Then I constructed a circle to each side of the median, such that the circle is a set of all points which are the vertices of angle $\alpha$ and $\beta$ above the given segment (the median) respectively. Now, if I only could create a line segment $AB$, such that it passes the point $C_1$ (the endpoint of the median and the middle point of side $AB$), each endpoint is on a different circle (to get the correct angles), and the line segments $AC_1$ and $C_1B$ are the same length ($C_1$ is the endpoint of a median), I would be done. Any ideas?

Answer 1

So, based on the suggestion of @Blue, the point is to find the midpoint of the centers of the circles $S$, construct a line segment $SC_1$, and then a line perpendicular to $SC_1$ passing through $C_1$. The two intersections with the circles are our points $A$ and $B$.

Answer 2

You can also employ this approach: draw a triangle $ABC$ with $\widehat{A}=\alpha$ and $\widehat{B}=\beta$, consider the length of the median $MA$ and apply a homotethy to $ABC$ in such a way that the length of $MA$ becomes the wanted length.