2 Bulgarian Winter Tournament

Question

Let ABC be a triangle, satisfying 2AC = AB + BC. If O and I are its circumcenter and incenter, show that OIB = 90^. (Konstantin Delchev)

Accepted Answer

Let D = BI (ABC). We apply Ptolemy's theorem AB DC + BC AD = AC BD which implies BD = 2DA. From AD = DI = DC it yields I is midpoint of BD so OIB = 90^.