MathLabs

第5题

设 ABCDEF ABCDEF be a convex hexag上使得 AB∥DE AB\parallel DE , BC∥EF BC\parallel EF , 且 CD∥FA CD\parallel FA . 设 RA,RC,RE R_A,R_C,R_E be circumradii 的三角形s FAB FAB , BCD BCD , DEF DEF , respectively, 且let p p be perimeter 的hexagon. 证明 RA+RC+RE≥p/2 R_A+R_C+R_E\ge p/2.
第 2/4 步：Project on到a perpendicular rect角
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通俗地说

Project on到a perpendicular rect角

2BF≥ABsin⁡B+AFsin⁡F+CDsin⁡C+DEsin⁡E2BF\ge AB\sin B+AF\sin F+CD\sin C+DE\sin E
详细分析

译文：Extend BC BC 且 FE FE , 且draw through A,D A,D perpendiculars 到them. The resulting rect角 has two relevant widths bounded 由 BF BF 译文：; resolving AB,AF,CD,DE AB,AF,CD,DE in到perpendicular projections gives displayed inequality. Cyclically analogous inequalities hold 对于 BD BD 且 FD FD .