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Radical axes of the circumcircles of the pairs of triangles

Source: Vietnam TST 1995, Problem 1

July 27, 2008
geometrycircumcirclegeometry unsolved

Problem Statement

Let be given a triangle ABC ABC with BC \equal{} a, CA \equal{} b, AB \equal{} c. Six distinct points A1 A_1, A2 A_2, B1 B_1, B2 B_2, C1 C_1, C2 C_2 not coinciding with A A, B B, C C are chosen so that A1 A_1, A2 A_2 lie on line BC BC; B1 B_1, B2 B_2 lie on CA CA and C1 C_1, C2 C_2 lie on AB AB. Let α \alpha, β \beta, γ \gamma three real numbers satisfy \overrightarrow{A_1A_2} \equal{} \frac {\alpha}{a}\overrightarrow{BC}, \overrightarrow{B_1B_2} \equal{} \frac {\beta}{b}\overrightarrow{CA}, \overrightarrow{C_1C_2} \equal{} \frac {\gamma}{c}\overrightarrow{AB}. Let dA d_A, dB d_B, dC d_C be respectively the radical axes of the circumcircles of the pairs of triangles AB1C1 AB_1C_1 and AB2C2 AB_2C_2; BC1A1 BC_1A_1 and BC2A2 BC_2A_2; CA1B1 CA_1B_1 and CA2B2 CA_2B_2. Prove that dA d_A, dB d_B and dC d_C are concurrent if and only if \alpha a \plus{} \beta b \plus{} \gamma c \neq 0.
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