So much generalized?!

Let $b_0$ be the side of the largest triangle

$b_1$ be the side of the 12 sq. unit triangle

$b_2$ be the side of the 27 sq. unit triangle

$b_3$ be the side of the 75 sq. unit triangle

The four triangles are similar to each other since they have parallel sides.

Say $A_{largest triangle} = \frac{1}{2}a_0b_0\sin C$

by sine law

$\frac{a_0}{\sin A} = \frac{b_0}{\sin B}$

substituting:

$A_{largest triangle} = \frac{b_0^2\sin A \sin C}{2\sin B}$

$b_0^2 = \frac{2A_{largest triangle}\sin B}{\sin A\sin C}$

$b_1^2 = \frac{2(12)\sin B}{\sin A\sin C}$

$b_2^2 = \frac{2(27)\sin B}{\sin A\sin C}$

$b_3^2 = \frac{2(75)\sin B}{\sin A\sin C}$

$b_0 = b_1 + b_2 + b_3$

$\sqrt{\frac{2A_{largest triangle}\sin B}{\sin A\sin C}} = \sqrt{\frac{2(12)\sin B}{\sin A\sin C}} + \sqrt{\frac{2(27)\sin B}{\sin A\sin C}} + \sqrt{\frac{2(75)\sin B}{\sin A\sin C}}$

Simplifying:

$\sqrt{2A_{largest triangle}} = \sqrt{2(12)} + \sqrt{2(27)} + \sqrt{2(75)}$

$\boxed{A_{largest triangle} = 300}$

Because we know that the drawn line are parallel to the sides of the triangle, all of the triangles are similar. The ratio of the areas is the square of the ratio of the sides for similar figures, so the ratio of the area of the triangle with area 12 is the square of the ratio of the side length of one of its sides to the corresponding side on the large triangle.

Because of the way the lines are constructed, each of the triangles has one vertex inside the triangle (where they all meet) and two vertices on a side of the large triangle. From those two vertices of the triangle with area 12, we construct lines parallel to the other side of the large triangle. This creates 2 more triangles. These triangles are congruent to the other two triangles by ASA (we have the side because the fact that the lines are parallel tells us that the 3 parts that are not triangles are parallelograms).

Therefore, the ratio of the side length of the triangle with area 12 to the side length of the big triangle is $\sqrt{12}:\sqrt{12}+\sqrt{27}+\sqrt{75}=2\sqrt{3}+3\sqrt{3}+5\sqrt{3}=10\sqrt{3}$ . Therefore, the ratio of the areas is $12:100*3=300$ . Because we have that the area of the triangle is 12, the area of the big triangle is $\boxed{300}$ .