AUSTRIAN-POLISH 2003 Problem 3

$ABC$ is a triangle. Take $a = BC$ etc as usual. Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$ . Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$ , and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$ . Show that if $a' = BT_5, b' = CT_1, c'=AT_3$ , then there is a triangle $A'B'C'$ with sides $a', b', c'$ ( $a' = B'C$ ' etc). In the same way we take points $T_i'$ on the sides of $A'B'C'$ and put $a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'$ . Show that there is a triangle $A'' B'' C''$ with sides $a'' b'' , c''$ and that it is similar to $ABC$ . Find $a'' /a$ .