SuccessiveArcsPath

Successive arcs path in a quadrangle

Consider a quadrangle p = ABCD. Start with an arbitrary point E on side AD 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 = EFG..., closing back to the start point E (arc-polygon-1) only in the case p is circumscriptible on a circle. The reason for that is explained in the file RotationsOnQuadrangleVertices.html . There it is shown that comming back to point I, on the same side with the start-point E, defines a segment (EI) equal to the translation-vector (v), which represents the composition of the four rotations about the vertices A, B, C and D by the angles of the quadrilateral at these vertices.

Analogous behaviour show the arc-polygons inscribed in more general polygons with more the four sides. The corresponding translation (EI) is zero, when the polygon is circumscriptible on a circle. Again there is a difference between odd- and pair- sided polygons. The discussion for this subject starts in the file SuccessiveArcs.html .

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