Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$ 
Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$

I start with the median $AM$. Since $\angle BAM, \angle CAM$ are known I can construct them. So I have point $A$, line segment $AM$ and $2$ rays starting from A where $\angle BAM, \angle CAM$ are those from the hypothesis. If only I could find a way to construct a line where $BM = MC$ I would be done. I can't figure out how to do that. 


The black lines are your dates. at first we draw green lines which are parallels to the rays in the point $M$, so we get the points $J,K$. with center in $J$ and radio $AJ$ we draw the red circle. similar with the point $K$ so we get the purple points $B,C$ and the segment $BC$ is your goal.
