Τρίτη, 17 Οκτωβρίου 2017

2003 IMO Shortlist G4

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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