$ABCDE$ is a regular pentagon. Determine the smallest value of the expression \[\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},\] where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.
Source: Czech-Polish-Slovak 2008, P5
Tags: geometry proposed, geometry
$ABCDE$ is a regular pentagon. Determine the smallest value of the expression \[\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},\] where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.