Point in a Regular Pentagon

This applet that deals with a regular pentagon is an extension of two earlier applets - "Point in an Equilateral Triangle" and "Point in a Square". After exploring the earlier applets as well as this one, what can you say about sum of the distances from a point in the interior of the pentagon to each of the sides? Why is this true? Under what circumstances can you form a new pentagon using the five distances to the sides? Can a similar result be obtained for any regular polygon with an odd number of sides? What about polygons with an even number of sides?