Problem 3.   Let D be an interior point of the acute triangle ABC with AB > AC so that ∠DAB = ∠CAD.  The point E on the segment AC satisfies ∠ADE = ∠BCD, the point  F  on the segment AB satisfies ∠FDA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O₁  and O₂ be the circumcentres of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O₁O₂  are concurrent.

proposed by Mykhailo Shtandenko, Ukraine