Let
![ABCDEF](/media/m/9/f/e/9fe205b534135e3a700ffb54d8b96cb0.png)
be a convex hexagon such that
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
is parallel to
![DE](/media/m/a/c/d/acdf3f4d3c794d9a897484e9d216f5ec.png)
,
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
is parallel to
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
, and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
is parallel to
![FA](/media/m/2/d/7/2d713d54952ae605212ea2176cb92816.png)
. Let
![R_{A},R_{C},R_{E}](/media/m/e/6/c/e6ca863424fd39672aceca039cb2c30e.png)
denote the circumradii of triangles
![FAB,BCD,DEF](/media/m/7/4/d/74d6b14c0ee4bbe99fe3fe2291bb75ea.png)
, respectively, and let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
denote the perimeter of the hexagon. Prove that
%V0
Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_{A},R_{C},R_{E}$ denote the circumradii of triangles $FAB,BCD,DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that
$$R_{A} + R_{C} + R_{E}\geq \frac {P}{2}.$$