MathLabs

Problem 1

Let ABCABC be a triangle with ∠BAC≠90∘\angle BAC\ne90^\circ. Let OO be the circumcenter of triangle ABCABC and let Γ\Gamma be the circumcircle of triangle BOCBOC. Suppose that Γ\Gamma intersects segment ABAB at P≠BP\ne B and segment ACAC at Q≠CQ\ne C. Let ONON be a diameter of Γ\Gamma. Prove that quadrilateral APNQAPNQ is a parallelogram.
Step 6 of 6: Conclusion
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AP∥QN,AQ∥PNAP\parallel QN,\quad AQ\parallel PN
Detailed analysis

Both pairs of opposite sides of quadrilateral APNQAPNQ are parallel; therefore it is a parallelogram.