Step 1: Use the meeting point of medians.
The two medians given are through B and C. Where any two medians meet is the centroid $G$ of the triangle. So first we find $G$.
Step 2: Solve the two median lines.
The medians are $x+y=5$ and $x=4$. Put $x=4$ into the first: $4+y=5$, so $y=1$. \[ G=(4,1) \]
Step 3: Find vertex B.
B lies on the median $x+y=5$, so write $B=(x_2,\,5-x_2)$. The centroid is the average of the three vertices. Using the $x$-values with $A=(1,2)$ and $C$ on $x=4$: \[ \frac{1+x_2+4}{3}=4 \implies x_2=7 \] Then $y_2=5-7=-2$, so $B=(7,-2)$.
Step 4: Find vertex C.
C lies on $x=4$, so $C=(4,y_3)$. Using the $y$-values in the centroid: \[ \frac{2+(-2)+y_3}{3}=1 \implies y_3=3 \] So $C=(4,3)$.
Step 5: Apply the area formula.
With $A(1,2)$, $B(7,-2)$, $C(4,3)$: \[ \text{Area}=\tfrac{1}{2}\,|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)| \]
Step 6: Plug in and finish.
\[ =\tfrac{1}{2}\,|1(-2-3)+7(3-2)+4(2+2)|=\tfrac{1}{2}\,|-5+7+16|=\tfrac{1}{2}(18) \] \[ \boxed{9} \]