Geometry · Trapezoids · Angle bisectors · Cyclic quadrilaterals · Pencils of lines
Let \(X\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\) (\(AB \parallel CD\)). Prove that if
\[ \angle ADX = \angle BCX, \]
then the bisectors of the angles \(\angle ADX\), \(\angle DXC\) and \(\angle XCB\) belong to a single pencil of lines, that is, either all three pass through one point or all three are parallel.
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Serbian Municipal Competition 2024, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source