15. In $\triangle A B C$, $\angle C=30^{\circ}, B M$ is a median, $A C=2 a$. If $\triangle A B C$ is folded along $B M$, then the overlapping area of the two smaller $\triangle A B M$ and $\triangle B C M$ is exactly $\frac{1}{4}$ of the area of $\triangle A B C$. Try to find the area of $\triangle A B C$ (expressed in terms of $a$).
15. Based on the size relationship between $\angle A B M$ and $\angle C B M$, there are three scenarios after folding.
(1) $\angle A B M > \angle C B M$.
Suppose $B C$ is folded to $B C^{\prime}$ (as shown in Figure 10), intersecting $A C$ at point $D$. Then
$$
\begin{array}{c}
S_{\triangle B D M} = \frac{1}{4} S_{\triangle A B C} \\
= \frac{1}{2} S_{\triangle A B M} \\
= \frac{1}{2} S_{\triangle C B M} = \frac{1}{2} S_{\triangle C B M}.
\end{array}
$$

Thus, $A D = M D$, $B D = C^{\prime} D$, and $\triangle A B D \cong \triangle M C^{\prime} D$.
Therefore, $A B = C^{\prime} M = C M = A M$.
Draw a perpendicular line from $A$ to $B C$, with the foot of the perpendicular being $B^{\prime}$. Since $\angle C = 30^{\circ}$, we have $A B^{\prime} = \frac{1}{2} A C = A M = A B$.
Thus, point $B$ coincides with $B^{\prime}$.
$$
\begin{array}{l}
\angle A B C = 90^{\circ}, A B = \frac{1}{2} A C = a, B C = \sqrt{3} a, \\
S_{\triangle A B C} = \frac{\sqrt{3}}{2} a^{2}.
\end{array}
$$
(2) $\angle A B M = \angle C B M$.

Then $\triangle B C M$ is folded to $\triangle B A M$, which clearly does not meet the problem's requirements.
(3) $\angle A B M < \angle C B M$.

Suppose $M C$ is folded to $M C^{\prime}$ (as shown in Figure 11), intersecting $A B$ at point $D$. Using the same proof method as in (1), we get $A D = B D$,
$$
\begin{array}{l}
M D = C^{\prime} D, \\
\triangle A M D \cong \triangle B C^{\prime} D.
\end{array}
$$

Thus, $B C^{\prime} = A M = B C = M C$.
Draw a perpendicular line from $B$ to $A C$, with the foot of the perpendicular being $E$.
Since $\angle C = 30^{\circ}$, we have $B E = \frac{1}{2} B C = \frac{1}{2} a$.
Therefore, $S_{\triangle A B C} = \frac{1}{2} A C \cdot B E = \frac{a^{2}}{2}$.