Geometry · Isosceles triangle · Angle bisector · Similar triangles · Excircle
Let \(ABC\) be an isosceles triangle with apex \(C\). Points \(D\) and \(E\) lie on the sides \(AC\) and \(BC\) respectively, and the bisector of the angle \(\angle DEB\) and the bisector of the angle \(\angle ADE\) meet at a point \(F\) that lies on the side \(AB\). Prove that \(F\) is the midpoint of \(AB\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2010, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source