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Jeremy Dunn
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Jan1711, 11:48 PM (EST) 

"Regular Polygon problem"

In doing some experimental drawing in AutoCAD I ran across what appear to be some interesting relations that appear to be true but which I am unable to prove regarding regular polygons. First, draw a regular polygon of 5 or more sides. Now pick a vertex and draw all rays going from that vertex to all of the other vertices, this divides the polygon into a series of triangles. The following seems to be true: 1. A circle passes through the incenters of all the triangles. 2. A circle passes through the centroids of all the triangles. 3. If you draw the incircles inside all of the triangles a circle can be drawn tangent to the outside of all of them and tangent to the circle passing through the vertices of the polygon. So my basic questions are 1. Are these relations true or just very close on my CAD system? 2. If true what are the formulas to calculate the radii of these circles? 3. How does one locate the centers of these circles? I suggest that it would be helpful to express distances in terms of setting the circumradius of the polygon equal to 1. Easy to state, much harder to solve! I suspect there are some interesting relations to be found here. 

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