Geometry · Tangent lines · Cyclic quadrilateral · Concyclicity · Angle chasing
In the plane, two circles \(k_1\) and \(k_2\) and a line \(p\) are given. The line \(p\) cuts \(k_1\) at the points \(A\) and \(B\), and it cuts \(k_2\) at the points \(C\) and \(D\). Each of the two tangents of \(k_1\) at \(A\) and at \(B\) is intersected with each of the two tangents of \(k_2\) at \(C\) and at \(D\); the four points obtained in this way are \(K\), \(L\), \(M\) and \(N\).
Prove that \(K\), \(L\), \(M\) and \(N\) lie on one circle.
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Serbian Regional Competition 2011, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source