IMO 1984 LL ROM48

Let ABC be a triangle with interior angle bisectors AA1,

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IMO 1984 LL ROM48

Origin: ROM

Problem

Let ABC be a triangle with interior angle bisectors AA1, BB1, CC1 and incenter I. If \sigma[IA1B] + \sigma[IB1C] + \sigma[IC1A] = 1 2\sigma[ABC], where \sigma[ABC] denotes the area of ABC, show that ABC is isosceles.