Geometry Problem 1142: Cyclic Quadrilateral, Isogonal Lines, Circle, Center, Radius, Perpendicular Bisector, Circumcenter. Level: High School, SAT Prep, Honors Geometry, College, Math Education

The infographic below shows a cyclic quadrilateral ABCD. E is a point inside triangle CFG so that angle BAC = angle DAE (AC and AE are isogonal lines with respect to angle BAC). The perpendicular bisector of CE intersects AB and AD at O₁ and O₂, respectively. Circle of center O₁ and radius O₁C intersects AB at F. Circle of center O₂ and radius O₂C intersects AD at G. Prove that the circumcenter O₃ of triangle EFG lies on AE.
 

See also: Geometry problem 1142, Solution sent by Stan & Mihai Fulger