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Let \Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4 be distinct circles such that \Gamma_1, \Gamma_3 are externally tangent at P, and \Gamma_2, \Gamma_4 are externally tangent at the same point P. Suppose that \Gamma_1 and \Gamma_2; \Gamma_2 and \Gamma_3; \Gamma_3 and \Gamma_4; \Gamma_4 and \Gamma_1 meet at A, B, C, D, respectively, and that all these points are different from P. Prove that

\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.

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