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three circles given, tangent wanted

Source: Dutch IMO TST2 2011 p3

August 6, 2019
circlestangentgeometry

Problem Statement

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two intersecting circles with midpoints respectively O1O_1 and O2O_2, such that Γ2\Gamma_2 intersects the line segment O1O2O_1O_2 in a point AA. The intersection points of Γ1\Gamma_1 and Γ2\Gamma_2 are CC and DD. The line ADAD intersects Γ1\Gamma_1 a second time in SS. The line CSCS intersects O1O2O_1O_2 in FF. Let Γ3\Gamma_3 be the circumcircle of triangle ADAD. Let EE be the second intersection point of Γ1\Gamma_1 and Γ3\Gamma_3. Prove that O1EO_1E is tangent to Γ3\Gamma_3.
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