MathDB

48

Part of 1984 IMO Longlists

Problems(1)

Given equation regarding areas, prove triangle is isosceles

Source:

10/12/2010
Let ABCABC be a triangle with interior angle bisectors AA1,BB1,CC1AA_1, BB_1, CC_1 and incenter II. If σ[IA1B]+σ[IB1C]+σ[IC1A]=12σ[ABC]\sigma[IA_1B] + \sigma[IB_1C] + \sigma[IC_1A] = \frac{1}{2}\sigma[ABC], where σ[ABC]\sigma[ABC] denotes the area of ABCABC, show that ABCABC is isosceles.
geometryincenteranalytic geometryangle bisectorgeometry proposed