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Danube Mathematical Competition 2007 Problem 2

Source: inscribed quadrilateral and 4 other circles tangent to its sides

December 8, 2007
geometrycircumcircleparallelogramgeometry proposed

Problem Statement

Let ABCD ABCD be an inscribed quadrilateral and let E E be the midpoint of the diagonal BD BD. Let Γ1,Γ2,Γ3,Γ4 \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4 be the circumcircles of triangles AEB AEB, BEC BEC, CED CED and DEA DEA respectively. Prove that if Γ4 \Gamma_4 is tangent to the line CD CD, then Γ1,Γ2,Γ3 \Gamma_1,\Gamma_2,\Gamma_3 are tangent to the lines BC,AB,AD BC,AB,AD respectively.
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