Given a quadrangle ABCD which accepts an inscribed circle, consider the composition f = f4*f3*f2*f1, of the rotations about its vertices by the angles at those vertices (the inner, assumed counterclockwise oriented). Prove that f is the identity. Thus for every point O of the plane the successive rotations of O come back to close to a quadrangle at O (OUVW).
It is easy to show that the closed quadrangle (OUVW), starting at O, is circumsribable by a circle, when O is on some side of (ABCD). In addition, the circumscribing circle of that quadrangle is concentric with the incircle of (ABCD).
The same problem, for general quadrangles is handled in the file: RotationsOnQuadrangleVertices.html .
Problem: Find the convexity region of O. This means the locations of O s.t. the resulting quadrangle (OUVW) is convex. When O is on the arc viewing AD under the angle ang((A+D)/2), then the three consecutive points W, O and U are collinear. I call this condition [collinearity at O]. (More general points O s.t. ang(WOU) has a constant value are on arcs of circles passing through A, D). There are similar conditions for the other vertices i.e [collinearity at U (O,U,V collinear)] etc. For points O far out in the plane, (OUVW) is always convex. Thus the convexity is destroyed in some domains (intersections) of the circles, carrying the arcs of [collinearity conditions].
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Rotating about quadrangle vertices ![[0_0]](RotationsOnQuadrangleVerticesCircum_files/RotationsOnQuadrangleVerticesCircum_0_0_0.png)
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![[1_0]](RotationsOnQuadrangleVerticesCircum_files/RotationsOnQuadrangleVerticesCircum_0_1_0.png)
![[1_1]](RotationsOnQuadrangleVerticesCircum_files/RotationsOnQuadrangleVerticesCircum_0_1_1.png)