VN-TST 1995 Problem 1

Let be given a triangle $ABC$ with $BC = a$ , $CA = b$ , $AB = c$ . Six distinct points $A_1$ , $A_2$ , $B_1$ , $B_2$ , $C_1$ , $C_2$ not coinciding with $A$ , $B$ , $C$ are chosen so that $A_1$ , $A_2$ lie on line $BC$ ; $B_1$ , $B_2$ lie on $CA$ and $C_1$ , $C_2$ lie on $AB$ . Let $\alpha$ , $\beta$ , $\gamma$ three real numbers satisfy $\overrightarrow{A_1A_2} = \frac {\alpha}{a}\overrightarrow{BC}$ , $\overrightarrow{B_1B_2} = \frac {\beta}{b}\overrightarrow{CA}$ , $\overrightarrow{C_1C_2} = \frac {\gamma}{c}\overrightarrow{AB}$ . Let $d_A$ , $d_B$ , $d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $AB_1C_1$ and $AB_2C_2$ ; $BC_1A_1$ and $BC_2A_2$ ; $CA_1B_1$ and $CA_2B_2$ . Prove that $d_A$ , $d_B$ and $d_C$ are concurrent if and only if $\alpha a + \beta b + \gamma c \neq 0$ .