Track / Stage 6 / 137 of 400 #1137 of 1964

Problem 1137

National Olympiad, first round

Geometry Difficulty 6.1 Prove it

Circle $O$ is inscribed in quadrilateral $A B C D$ , touching the non-parallel sides $B C, A D$ at points $E, F$ , respectively. Let the line $A O$ intersect the segment $E F$ at point $K$ , and the line $D O$ intersect the segment $E F$ at point $N$ . The line $B K$ intersects the line $C N$ at point $M$ . Prove: $O, K, M$ and $N$ are concyclic.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.