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Show that exceptional points lie on the same circle

Source: APMO 2009 Q.3

March 13, 2009
radical axisgeometry unsolvedgeometry

Problem Statement

Let three circles Γ1,Γ2,Γ3 \Gamma_1, \Gamma_2, \Gamma_3, which are non-overlapping and mutually external, be given in the plane. For each point P P in the plane, outside the three circles, construct six points A1,B1,A2,B2,A3,B3 A_1, B_1, A_2, B_2, A_3, B_3 as follows: For each i \equal{} 1, 2, 3, Ai,Bi A_i, B_i are distinct points on the circle Γi \Gamma_i such that the lines PAi PA_i and PBi PB_i are both tangents to Γi \Gamma_i. Call the point P P exceptional if, from the construction, three lines A1B1,A2B2,A3B3 A_1B_1, A_2 B_2, A_3 B_3 are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.
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