Constructing a triangle with given $\alpha, a, m_b$.

Question

I've to construct a triangle with given $\alpha,a,m_b$ where $m_b$ is a median to $b$.

Here is my attempt. $c_1$ is a set of all possible points $A$ (e.g. we see line $BC$ at $\alpha$). $c_2$ is the set of all possible points $S_b$ which are the middle of line $AC$. Now, I've to find the point $S_b$ on $c_2$ such that $|SA|=|SC|$ and also $A\in c_1$. How do I achieve that?

Answer 1

Let $M$ be a midpoint of $AC$. Draw segment $MB =m_b$ and a circle $S$ with chord $MB$ and central angle over that chord $2\alpha$. Then reflect this circle across point $M$ and we get new circle $S'$. Since $C$ is reflection of $A$ across $M$ point $C$ must lie on $S'$. If we draw now a circle with center at $B$ and radius $a$ you will get in intersection with $S'$ point $C$. ...