MathLabs

Problem 5

Let ABCDEF ABCDEF be a convex hexagon such that AB∥DE AB\parallel DE , BC∥EF BC\parallel EF , and CD∥FA CD\parallel FA . Let RA,RC,RE R_A,R_C,R_E be the circumradii of triangles FAB FAB , BCD BCD , DEF DEF , respectively, and let p p be the perimeter of the hexagon. Prove that RA+RC+RE≥p/2 R_A+R_C+R_E\ge p/2.
Step 2 of 4: Project onto a perpendicular rectangle
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In plain words

Project onto a perpendicular rectangle

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

Extend BC BC and FE FE , and draw through A,D A,D perpendiculars to them. The resulting rectangle has the two relevant widths bounded by BF BF ; resolving AB,AF,CD,DE AB,AF,CD,DE into perpendicular projections gives the displayed inequality. Cyclically analogous inequalities hold for BD BD and FD FD .