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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 4 of 5: Identify the equal powers
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OOi2−ρ2=OiXi⋅OiP=ri2OO_i^2-\rho^2=O_iX_i\cdot O_iP=r_i^2
Detailed analysis

Because X_i and P are intersections of the line O_iP with Ω, O_iX_i·O_iP is the power of O_i with respect to Ω. Thus OO_i^2−ρ^2=r_i^2, or equivalently OO_i^2−r_i^2=ρ^2.