SuccessiveArcsPenta

Successive arcs polygon

Consider a circumscriptible pentagon p = ABCDE. Start with an arbitrary point F on side ED 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 = FGHIJKLMNO, closing back to the start point F (arc-polygon-1). In the same vein, one constructs the arc-polygon p2 = PQRST 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 {PF = PK = GQ = QL = ... }, and the vertices of the first arc-polygon p1 are on a circle, concentric to the incircle of the original pentagon p.

Analogous properties are valid for circumscriptible polygons with an even number of sides. For such polygons the corresponding arc-polygon p1, closes after visiting each side once and not two times as in the odd-number of sides case. Thus, the resulting arc-polygon p1 has equal number of vertices with the original polygon. An example, for a circumscriptible hexagon and suggestions for the proofs are given in SuccessiveArcsHex.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/