  Viviani's theorem states that the sum of distances from any point inside an
equilateral triangle to its sides is constant. We consider extensions of the
theorem and show that any convex polygon can be divided into parallel segments
such that the sum of the distances of the points to the sides on each segment
is constant. A polygon possesses the CVS property if the sum of the distances
from any inner point to its sides is constant. An amazing result, concerning
the converse of Viviani's theorem is deduced; Three non-collinear points which
have equal sum of distances to the sides inside a convex polygon, is sufficient
for possessing the CVS property. For concave polygons the situation is quite
different, while for polyhedra analogous results are deduced.
