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].

Produced with EucliDraw© |