Answer: 315

Small Hint:

Call the upper-right unlabeled area $x$ and the upper-left unlabeled area $y$

Big Hint:

Use equal-altitude area ratios along the sides, together with Ceva’s theorem

Solution:

Let the unlabeled upper-right and upper-left areas be $x$ and $y,$ respectively. The three side-division ratios and Ceva’s theorem give $$\frac43\cdot\frac{35}{x}\cdot\frac{84}{y}=1,$$ so $xy=3920.$

The cevian from $A$ meets $BC.$ The ratio of the two segments of $BC$ is $\frac{35}{x}.$ Computing the same ratio from the two large triangles with vertex $A$ gives $$\frac{35}{x}=\frac{40+30+35}{x+y+84},$$ so $y+84=2x.$ Solving with $xy=3920$ yields $x=70$ and $y=56.$ Therefore $$\begin{aligned} [ABC]&=84+70+35\\ &\quad{}+30+40+56=315. \end{aligned}$$