SuccessiveArcsHex

Hexagonal arc-polygon

Consider a circumscriptible hexagon p = ABCDEF. Start with an arbitrary point G on side EF and draw arcs, centered at the vertices and end-points on adjacent sides, so that the start-point of the next is the end-point of the previous. The construction results in an arc-polygon p1 = GJKNOR, closing back to the start point G (arc-polygon-1).
In the same vein, one constructs the arc-polygon p2 = HILMPQ of the contact points of the circumscriptible hexagon with its incircle (arc-polygon-2).
The two arc polygons are indeed closed, their vertices on the same side define equal segments, and the vertices of the first arc-polygon p1 are on a circle, concentric to the incircle of the original polygon p.

The arc-polygon p2, of the contact points of the circumscriptible hexagon with its incircle (arc-polygon-2) is obviously closed, since the tangents from a point to a circle are equal. The segments, {HG, IJ, LK, ... } are equal, since they are pairwise defined by concentric circles. The first arc-polygon p1 is, then, easily seen, to be closed. Since the equal segments, mentioned above, are also tangent to the incircle, their end-points, which are the vertices of p1, are on a circle concentric to the incircle of the original polygon. For a continuation of the discussion, look at the file SuccessiveArcsHex2.html .

The subject is related to the composition of rotations about the vertices of a polygon. The vertices of the arc-polygon p1 constitute an orbit of the group generated by these rotations. Look at the file RotationsOnQuadrangleVertices.html , for a discussion of these matters.

http://users.math.uoc.gr/~pamfilos/