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AA_0, BB_0,IC_1 concurrent when for areas when (ABC) = 2 (A_1B_1C)

Source: 2010 Balkan Shortlist G3 BMO

April 4, 2020
concurrencyconcurrentarea of a triangleareasgeometryincenterincircle

Problem Statement

The incircle of a triangle A0B0C0A_0B_0C_0 touches the sides B0C0,C0A0,A0B0B_0C_0,C_0A_0,A_0B_0 at the points A,B,CA,B,C respectively, and the incircle of the triangle ABCABC with incenter I I touches the sides BC,CA,ABBC,CA, AB at the points A1,B1,C1A_1, B_1,C_1, respectively. Let σ(ABC)\sigma(ABC) and σ(A1B1C)\sigma(A_1B_1C) be the areas of the triangles ABCABC and A1B1CA_1B_1C respectively. Show that if σ(ABC)=2σ(A1B1C)\sigma(ABC) = 2 \sigma(A_1B_1C) , then the lines AA0,BB0,IC1AA_0, BB_0,IC_1 pass through a common point .
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