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AA_1xAA_2 + BB_1xBB_2 + CC_1xCC_2 >=1/9 (AB^2 + BC^2 + CA^2)

Source: 2011 Balkan Shortlist G4 BMO

April 3, 2020
geometric inequalitygeometryincircle

Problem Statement

Given a triangle ABCABC, the line parallel to the side BCBC and tangent to the incircle of the triangle meets the sides ABAB and ACAC at the points A1A_1 and A2A_2 , the points B1,B2B_1, B_2 and C1,C2C_1, C_2 are de ned similarly. Show that AA1⋅AA2+BB1⋅BB2+CC1⋅CC2≥19(AB2+BC2+CA2)AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)
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