Geometry · Incircle · Contact triangle · Cyclic quadrilaterals · Angle chasing
Problem 3, 2018
The circle inscribed in triangle \(ABC\) touches the sides \(BC\), \(CA\) and \(AB\) at the points \(D\), \(E\) and \(F\) respectively. A point \(K\) lies on the same side of the line \(EF\) as the vertex \(A\), and satisfies
\[ \angle KFE = \angle ACB \qquad \text{and} \qquad \angle KEF = \angle ABC . \]
Prove that \(KD \perp BC\).
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Serbian Regional Competition 2018, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source