Let's construct the triangle $ABC$, given the side $BC=a$, the median $BB_{1}=s_{b}$ (where $B_{1}$ is the midpoint of side $AC$), and the angle $ABB_{1}=\beta_{2}$.
Consider the problem solved and look at the reflection of triangle $B C B_{1}$ over $B_{1}$ to get the reflection of $D A B_{1}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_1f733c26b726bb5b67abg-1.jpg?height=361&width=509&top_left_y=188&top_left_x=797)

Thus, in triangle $A B D$, we know the sides $B D=2 s_{b}$ and $D A=C B=a$, and the angle $\angle A B D=\beta_{2}$ opposite to the latter. Constructing this triangle is a basic problem; then we can obtain $C$ by reflecting $A$ over $B_{1}$.

For $a \geq 2 s_{b}$, the construction can be carried out uniquely; however, for $a<2 s_{b}$, the number of solutions can be 2, 1, or 0.

Lörincz Csaba (Orosháza, Táncsics M. g. I. o. t.)

Remark. Essentially, we solve the problem in the same way if we first construct triangle $A_{1} B_{1} B$, where $A_{1}$ is the midpoint of side $B C$.