Geometry · Incircle · Tangent segments · Angle bisector · Perpendicular bisector · Circumcentre · Isosceles triangle
The incircle of a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(D\) and \(E\) respectively. Let \(F\) be any point of the side \(AB\) lying between \(A\) and \(D\), and let \(G\) be any point of the side \(AC\) lying between \(E\) and \(C\). On the segment \(FG\) take the point \(T\) for which the triangle \(EGT\) is isosceles with apex \(G\), that is \(|GE| = |GT|\). Prove that the circumcentre of the triangle \(DET\) is the same point as the incentre of the triangle \(AFG\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2006, 1. letnik, category A, problem 4. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source