Geometry · Convex quadrilateral · Equilateral triangle · Concyclic points · Inscribed angle · Angle bisector · Concurrency
A convex quadrilateral \(ABCD\) satisfies
Points \(P\) and \(Q\) are chosen so that the triangles \(ACP\) and \(BDQ\) are equilateral, with \(P\) lying in the half-plane bounded by the line \(AC\) that does not contain \(B\), and \(Q\) in the half-plane bounded by the line \(BD\) that does not contain \(A\). Prove that the lines \(PQ\), \(AD\) and \(BC\) all pass through one point.
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Serbian Municipal Competition 2013, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source