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R.I.P synthetic geo

Source: Iran MO 3rd round 2023 ,Day 1 P1

August 16, 2023
geometry

Problem Statement

In triangle △ABC\triangle ABC , M,NM, N are midpoints of AC,ABAC,AB respectively. Assume that BM,CNBM,CN cuts (ABC)(ABC) at M′,N′M',N' respectively. Let XX be on the extention of BCBC from BB st ∠N′XB=∠ACN\angle N'XB=\angle ACN. And define YY similarly on the extention of BCBC from CC. Prove that AX=AYAX=AY.
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