emat6690
Written by Sunny Yoon

 

 

Polygonal Path

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. . .

copywilson

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.

 


quad-convex

concave - quad

 

After these two examples, I wanted to explore if there exists a difference between a convex/concave polygon.

 
Convex Polygon
Concave Polygon
Pentagon
pentagon-convex
pentagon-concave
Hexagon
hexagon-convex
hexagon-concave
Heptagon
heptagon-convex
heptagon-concave
Octagon octagon-convex octagon-concave

 

convex-short

 

longlength-convex

 

explain-concave

 

GSP file

 


 

Now, I decided to investigate regular polygon and using the center of the polygon as point S.

Square square
Pentagon pentagon
Hexagon hexagon
Octogon octogon

explain-regular

 

GSP file


 

Now, I'm going to investigate what happens when a polygon is a triangle and point S is the center of a triangle.

Orthocenter
orthocenter
Centroid
centroid

Circumcenter

circumcenter
Incenter
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 obtuse
Centroid obtuse
Circumcenter ob
Incenter ob

 

What is the polygonal path when it is a right triangle?

Orthocenter r
Centroid r
Circumcenter r
Incenter r