RotationsOnQuadrangleVertices

Rotating about quadrangle vertices

Given an oriented quadrangle ABCD consider the composition f = f4*f3*f2*f1 of the four rotations on its vertices by the angles at those vertices. f is a translation. This is immediately seen by reducing the rotations to products of reflexions, using the bisectors of the quadrangle. These bisectors form another quadrangle, which is always circular one.
The requested f becomes the product of reflections f = g*e*c*a. For circular quadrangles this is always a translation. In particular this translation preserves the points of the first side AD, hence it is parallel to it (we rotate counterclockwise successively about A, B, C and D).

In particular, when the original quadrangle ABCD accepts an inscribed circle, then m = 0, and for every X, f(X) = X.
This problem is consdered in the file: RotationsOnQuadrangleVerticesCircum.html .

A related subject, resulting by replacing rotations with reflexions is handled in the file: ReflectingOnPolygonSides.html .

For an application on circular-arcs-polygons, inscribed in other polygons look at the file: SuccessiveArcsPath.html .

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