This is a Geometer's Sketpad animation where a point P is animated about a polygonal path. An envelope of circles is traced where P is the center of each circle and each circle passes through a fixed point S. Explore. . .
This problem is presented in Dr. Wilson's homepage for EMAT 6690.
I decided to investigate polygonal path using different polygons or use a fixed point S as a center of a triangle and see how an envelope of circles is traced around the polygon.
After these two examples, I wanted to explore if there exists a difference between a convex/concave polygon.
Convex Polygon 
Concave Polygon 

Pentagon  
Hexagon  
Heptagon  
Octagon 
Now, I decided to investigate regular polygon and using the center of the polygon as point S.
Square  
Pentagon  
Hexagon  
Octogon 
Now, I'm going to investigate what happens when a polygon is a triangle and point S is the center of a triangle.
Orthocenter 

Centroid 

Circumcenter 

Incenter 
I used the same triangle when I was finding the center of the triangle.
What is the polygonal path when it is an obtuse triangle?
orthocenter  

Centroid  
Circumcenter  
Incenter 
What is the polygonal path when it is a right triangle?
Orthocenter  

Centroid  
Circumcenter  
Incenter 