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 3 of 4: Use parallelism
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In plain words

Use parallelism

A=D,B=E,C=FA=D,\quad B=E,\quad C=F
Detailed analysis

Opposite sides are parallel, so the corresponding interior angles are equal: A=D A=D , B=E B=E , and C=F C=F . Divide the three projection inequalities by sin⁡A,sin⁡C,sin⁡E\sin A,\sin C,\sin E and add them.