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IMO ShortList 2003, geometry problem 4

Source: IMO ShortList 2003, geometry problem 4

October 4, 2004
geometryparallelogramhomothetyInversionIMO Shortlistgeometry solvedlengths

Problem Statement

Let Γ1\Gamma_1, Γ2\Gamma_2, Γ3\Gamma_3, Γ4\Gamma_4 be distinct circles such that Γ1\Gamma_1, Γ3\Gamma_3 are externally tangent at PP, and Γ2\Gamma_2, Γ4\Gamma_4 are externally tangent at the same point PP. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2; Γ2\Gamma_2 and Γ3\Gamma_3; Γ3\Gamma_3 and Γ4\Gamma_4; Γ4\Gamma_4 and Γ1\Gamma_1 meet at AA, BB, CC, DD, respectively, and that all these points are different from PP. Prove that AB⋅BCAD⋅DC=PB2PD2. \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
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