The following diagram depicts a Pass the Baton tournament
(Note: due to a merge, some of the answers below use this image instead.)
Each team consists of 6 members, 2 at the central point $E$, and one on each side of the quadrilateral, $a$, $b$, $c$, and $d$. The members of the race run to the sides in the following order: $E$ to side $a$, $a$ to $b$, $b$ back to $E$, $E$ to $c$, $c$ to $d$, and $d$ back to $E$.
The challenge for the teams is not only to run as fast as they can but to select the best route, by selecting the best points on $a$, $b$, $c$, and $d$ to have the relay runners stand.
What is this route?