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". You can adjust the size and orientation of the pentagon by dragging the small white dots. 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? Can you devise a way to deal with the cases in which the sum of the distances is undefined? What is the shape of the region for which the sum is fixed? 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?