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Part of 2018 Bosnia and Herzegovina EGMO TST

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Bosnia and Herzegovina EGMO TST 2018 Problem 3

Source: Bosnia and Herzegovina EGMO Team Selection Test 2018

9/19/2018
Let OO be a circumcenter of acute triangle ABCABC and let O1O_1 and O2O_2 be circumcenters of triangles OABOAB and OACOAC, respectively. Circumcircles of triangles OABOAB and OACOAC intersect side BCBC in points DD (D≠BD \neq B) and EE (E≠CE \neq C), respectively. Perpendicular bisector of side BCBC intersects side ACAC in point FF(F≠AF \neq A). Prove that circumcenter of triangle ADEADE lies on ACAC iff FF lies on line O1O2O_1O_2
geometrycircumcircleperpendicular bisector