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 |  |