Here are some things I don't understand? What does the problem mean by "bisectrices" (plural of bisectrix)? Does it mean "bisectors"? What kind of bisectors? Segment bisectors? (medians or perpendicular bisectors) Angle bisectors?

bisectrices mean Angle bisectors

Reflect on $AA'$. We know that the lines $AB,B'A'$ will reflect to the lines $AC,C'A'$, so $B'$ maps to $C'$ and hence $AB'=AC'$. Similarly, $BA'=BC'$ and $CA'=CB'$, so $A'B'C'$ is the contact triangle of $\triangle ABC$. This implies that say $\frac{AC'}{C'B}=\frac{s-a}{s-b}=\frac{b}{a}$, $a(s-a)=b(s-b)$, $ab+ac-a^2=ba+bc-b^2$, $b^2-a^2+ac-bc=0$, $(b-a)(b+a-c)=0$. By the Triangle inequality, $b+a-c>0$, giving $a=b$. Similarly, we get $b=c$ and $c=a$, so $\triangle ABC$ is indeed equilateral.

the bundle AB,AC,AA' and AE where AE is the ex-bisector of the angle A is harmonic. we assert that A' is midpoint of BC .If not then AE intersect BC at E therefore EBA'C harmonic and also EC'A"B' harmonic [A" the intersection of AA' with C'B'] thus A'E is the ex-bisector of <B'AC' but AE //A' E implies A in BC which is absurd.