Place $A B C$ in the coordinate plane so that $A=(0,6), B=(0,0), C=(8,0)$. Consider a point $P=(a, b)$ inside triangle $A B C$. Clearly, $d(P, A B)=a, d(P, B C)=b$. Now, we see that the area of triangle $A B C$ is $\frac{6 \cdot 8}{2}=24$, but may also be computed by summing the areas of triangles $P A B, P B C, P C A$. The area of triangle $P A B$ is $\frac{6 \cdot a}{2}=3 a$, and similarly the area of triangle $P B C$ is $4 b$. Thus, it follows easily that $d(P, C A)=\frac{24-3 a-4 b}{5}$. Now, we have $$(x, y)=\left(\frac{24}{5}-\frac{8}{5} a-\frac{4}{b} b, \frac{24}{5}-\frac{3}{5} a-\frac{9}{5} b\right)$$ The desired region is the set of $(x, y)$ obtained by those $(a, b)$ subject to the constraints $a \geq 0, b \geq$ $0,6 a+8 b \leq 48$. Consequently, our region is the triangle whose vertices are obtained by evaluating $(x, y)$ at the vertices $(a, b)$ of the triangle. To see this, let $f(a, b)$ output the corresponding $(x, y)$ according to the above. Then, we can write every point $P$ in $A B C$ as $P=m(0,0)+n(0,6)+p(8,0)$ for some $m+n+p=1$. Then, $f(P)=m f(0,0)+n f(0,6)+p f(8,0)=m\left(\frac{24}{5}, \frac{24}{5}\right)+n(-8,0)+p(0,-6)$, so $f(P)$ ranges over the triangle with those three vertices. Therefore, we need the area of the triangle with vertices $\left(\frac{24}{5}, \frac{24}{5}\right),(0,-6),(-8,0)$, which is easily computed (for example, using determinants) to be $\frac{288}{5}$.