I find a larger area than @Matt in a non-symmetric solution. A numerical approximation to that solution is:

     [[[-0.3204647107, -0.4111035999], [0.1858745389, -0.4555874153], [0.1345901718, 3.498839041]], 
      [[2.842082885, -1.172240373], [-0.31551567, -0.3800257369], [-0.2470601577, 0.2361920968]], 
      [[-2.635080319, -1.218751491], [0.2046624437, -0.3929906911], [0.1509108184, 0.2956681691]]]
This can then be turned in an exact solution by first viewing those decimals numbers as rational numbers an then rescaling each triangle to have exactly volume 1 and translating such that they exactly have the desired center. 
Doing all these calculations in $\mathbb{A}$ can be done, but the solutions have large minimial polynomials, too large to fit in the margin at least...
 
Then one can calculate the area of the intersection of these three triangles and gets $0.248595187378...$. 

As given above they approximate the required conditions to at least 10 decimal digits, so even without relying on exact calculations in $\mathbb{A}$, I'm pretty confident that this larger area is not just a numerical artifact.


Here's an image of the arrangement:

<img src="https://raw.githubusercontent.com/mo271/mo271.github.io/master/mo/406508/three_triangles.svg">


I found this solution trying a out random triangles. Of course this is not the optimal solution, I just wanted to point out that it might be a good idea to look into solution not assuming $D_3$ symmetry, but perhaps rather into solution that only have the $\mathbb{Z}/2$ symmetry of the original triangle, because the solution I found looks like it is close to a local maximum where the green and the orange triangle are mirror images of each other and the purple one is different. 

**Update:** Instead of doing random exploration, I used some gradient descent numerical optimization to look for more local optima. As suspected the solution with $\mathbb{Z}/2$-symmetry comes up. So even without imposing symmetry the following symmetry  
$${\\((a, b), (-a, b), (0, c)),\\\\      ((d, e), (f, g), (h, i)),\\\\            ((-d,e), (-f,g), (-h,i))\\}$$
Choosing approximately
$$ (a, b, c, d, e, f, g, h, i) = \\(-0.23003 , -0.571673,  3.775494,  2.701848, -0.885711,
       -0.229117, -0.55433, -0.19322 ,  0.123974)$$
gives an area of approximately $0.2517...$. This it what it looks like:
<img src="https://raw.githubusercontent.com/mo271/mo271.github.io/master/mo/406508/three_triangles_v2.svg">

I didn't prescribe the vertices of the green and yellow triangle to lie on the sides of the purple triangle. 
Looking at the numerical evindence, I would be surprised if the local optimum near the solution described above is not the *global* optimum. It might be possible to get exact coordinates for the local optimum.