Geometry · Circles · Inscribed angles · Common chord
Two circles \(k_{1}\) and \(k_{2}\) intersect at two distinct points, and \(AB\) is their common chord. A point \(P\) is chosen on \(k_{1}\) so that it lies outside \(k_{2}\). The lines \(PA\) and \(PB\) meet \(k_{2}\) a second time at points \(X\) and \(Y\), respectively. Show that the length of the segment \(XY\) does not depend on the choice of the point \(P\).
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Serbian Municipal Competition 2026, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source