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Tangencies in Quadrilateral

Source: Turkey JBMO TST 2015 P2

June 23, 2016
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral and let ω\omega be a circle tangent to the lines ABAB and BCBC at points AA and CC, respectively. ω\omega intersects the line segments ADAD and CDCD again at EE and FF, respectively, which are both different from DD. Let GG be the point of intersection of the lines AFAF and CECE. Given ∠ACB=∠GDC+∠ACE\angle ACB=\angle GDC+\angle ACE, prove that the line ADAD is tangent to th circumcircle of the triangle AGBAGB.
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