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concurrent wanted, intersecting circles related

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2004 Seniors p2

September 6, 2020
geometrycirclesconcurrencyconcurrentChampions Tournament

Problem Statement

Two different circles ω1\omega_1 ,ω2\omega_2, with centers O1,O2O_1, O_2 respectively intersect at the points A,BA, B. The line O1BO_1B intersects ω2\omega_2 at the point F(F≠B)F (F \ne B), and the line O2BO_2B intersects ω1\omega_1 at the point E(E≠B)E (E\ne B). A line was drawn through the point BB, parallel to the EFEF, which intersects ω1\omega_1 at the point M(M≠B)M (M \ne B), and ω2\omega_2 at the point N(N≠B)N (N\ne B). Prove that the lines ME,ABME, AB and NFNF intersect at one point.
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