Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b  \] In what case does the equality hold?
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   - If \(b \tan{\frac{w}{2}} = c\), then \(b = AK\), resulting in only one solution where \(\triangle ABC\) is a right triangle with \(\angle A = 90^\circ\).

\(\blacksquare\)

The final answer is \( \boxed{ b \tan{\dfrac{w}{2}} \leq c < b } \).