Consider a convex quadrangle p = (BCDE) and a point A inside it. Build the quadrangle q = (FGIH), having as vertices the reflections of A on the sides of p. The problem is to find the domain of convexity of q. This means to find all locations for A, for which the resulting q is convex.
In the figure the convexity region is the yellow region, bounded by two sides of the quadrangle and two arcs of circles, formed by the intersection points of opposite sides of p. When A is on one of these arcs, on BC say, then the reflections of A on the sides of triangle BCK are on a line (Steiner/Simson).
Look at Domain_of_Convexity.html , for a related figure. The generalization for (convex) polygons with more than four sides is obvious.
Look at Convexity_Domain_Entire.html , for the entire convexity domain, i.e. when A is not restricted to be inside the quadrangle.
![[alogo]](alogo.png){kind=small}
Domain of convexity for quadrangles ![[0_0]](Domain_of_Convexity_quad_files/Domain_of_Convexity_quad_0_0_0.png)
![[0_1]](Domain_of_Convexity_quad_files/Domain_of_Convexity_quad_0_0_1.png)