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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 5 of 5: Fix the circle
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Pow⁡Γ1(O)=Pow⁡Γ2(O)=Pow⁡Γ3(O)=ρ2\operatorname{Pow}_{\Gamma_1}(O)=\operatorname{Pow}_{\Gamma_2}(O)=\operatorname{Pow}_{\Gamma_3}(O)=\rho^2
Detailed analysis

The last equality holds for i=1,2,3, so O is the radical center of the three given circles and is independent of the exceptional point P. The common power ρ^2 is also fixed. Consequently every exceptional P lies on the one fixed circle centered at O with radius ρ.