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geometric inequality with side ratios

Source: Brazil TST 1999 Test 2 P2

April 30, 2021
geometric inequalityinequalitiesgeometryratio

Problem Statement

In a triangle ABCABC, the bisector of the angle at AA of a triangle ABCABC intersects the segment BCBC and the circumcircle of ABCABC at points A1A_1 and A2A_2, respectively. Points B1,B2,C1,C2B_1,B_2,C_1,C_2 are analogously defined. Prove that A1A2BA2+CA2+B1B2CB2+AB2+C1C2AC2+BC2≥34.\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.
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