MathLabs

Problem 3

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 PP in the plane, outside the three circles, construct six points A1,B1,A2,B2,A3,B3A_1,B_1,A_2,B_2,A_3,B_3 as follows: for each i=1,2,3i=1,2,3, Ai,BiA_i,B_i are distinct points on Γi\Gamma_i such that the lines PAiPA_i and PBiPB_i are both tangents to Γi\Gamma_i. Call PP exceptional if the three lines A1B1,A2B2,A3B3A_1B_1,A_2B_2,A_3B_3 are concurrent. Show that every exceptional point of the plane, if any exists, lies on the same circle.
Step 1 of 5: Build the diameter circle
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A1B1∩A2B2∩A3B3=Q,Ω=(PQ),O=center⁡(Ω)A_1B_1\cap A_2B_2\cap A_3B_3=Q,\quad \Omega=(PQ),\quad O=\operatorname{center}(\Omega)
Detailed analysis

Let P be exceptional and let Q be the common point of A_1B_1,A_2B_2,A_3B_3. Construct Ω with diameter PQ, center O, and radius ρ.