I find an area of $.122732$ for the intersection when each $T_i$ has a vertex at distance $2.23329$ from the center of $T$. This seems to be maximal among the symmetric options.

The gray triangle in the diagram is $T$, the black triangles are the $T_i$ (with their specified far vertices outside the area of the diagram), and the nonagonal blue region is the desired intersection.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/XrHcB.png