14. (1) In quadrilateral $A B M C$, applying the generalized Ptolemy's theorem (Ptolemy's inequality) we get
$$A M \cdot B C \leqslant B M \cdot A C + C M \cdot A B$$

In $\triangle A B C$, by the Law of Sines we have
$$A M \cdot 2 R \sin A \leqslant B M \cdot 2 R \sin B + C M \cdot 2 R \sin C$$

which simplifies to
$$A M \cdot \sin A \leqslant B M \cdot \sin B + C M \cdot \sin C$$
(2) From (1), we get
$$\begin{array}{l} A A_{1} \cdot \sin \alpha \leqslant A B_{1} \cdot \sin \beta + A C_{1} \cdot \sin \gamma \\ B B_{1} \cdot \sin \beta \leqslant B A_{1} \cdot \sin \alpha + B C_{1} \cdot \sin \gamma \\ C C_{1} \cdot \sin \gamma \leqslant C A_{1} \cdot \sin \alpha + C B_{1} \cdot \sin \beta \end{array}$$

Adding the three inequalities, we get
$$A A_{1} \cdot \sin \alpha + B B_{1} \sin \beta + C C_{1} \sin \gamma \leqslant B C \cdot \sin \alpha + C A \cdot \sin \beta + A B \cdot \sin \gamma$$