Step-by-step Solution:

Since the angle of elevation from the three vertices is the same, denoted by \( \alpha \), the tower must be equidistant from the three vertices, meaning the tower is at the circumcenter of the triangle. The distance of the circumcenter from any vertex is: \[ h \cot \alpha = R \quad \text{(circum-radius)}. \] We know the following relationship in a triangle: \[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2R}. \] Hence, \[ h \cot \alpha = \frac{a}{2 \sin A} \quad \Rightarrow \quad h = \frac{a}{2} \tan \alpha \csc A. \]