Geometry · Cyclic polygons · Midlines · Concyclicity · Inscribed angles
A pentagon \(ABCDE\) is inscribed in a circle. Let \(F\), \(G\), \(H\) and \(I\) be the midpoints of the segments \(BC\), \(CD\), \(DE\) and \(EA\) respectively. The lines \(FG\) and \(HI\) meet at the point \(J\), and the lines \(AC\) and \(HI\) meet at the point \(K\).
Prove that \(K\) lies on the circle circumscribed about the triangle \(FCJ\).
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Serbian Regional Competition (Okruzno takmicenje) 2019, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source