In a triangle $ABC$ with incircle $\omega$, excenters $I_A, I_B, I_C$, let $t_{AB}$ and $t_{AC}$ be the tangent lines to $\omega$ through $I_A$ that are closer to $B$ and $C$ respectively. Construct similarly $t_{BA}$, $t_{BC}$, $t_{CA}$, $t_{CB}$.
Define $D:=t_{BA} \cap t_{CA}$, $E:=t_{CB} \cap t_{AB}$, $F:=t_{AC} \cap t_{BC}$.
My question is:
is there a way to prove that lines $AD$, $BE$, $CF$ do NOT concur? I am particularly interested in a synthetic method that does not involve writing everything in trilinear coordinates and doing all the calculations.
is there a reason why, even though they do not concur, they seem to be very close (with respect to the size of $ABC$), which may come as a surprise given the "natural" construction we performed? Why this almost concurrence?
I draw a picture to show they are not concur. it is simply you may not go to such triangle which is a little hard to draw .