id: 05888378
dt: j
an: 2011c.00530
au: Abboud, Elias
ti: On Viviani’s theorem and its extensions.
so: Coll. Math. J. 41, No. 3, 203-211 (2010).
py: 2010
pu: Mathematical Association of America (MAA), Washington, D.C.
la: EN
cc: G45
ut: Viviani theorem; linear programming; convex polygons
ci:
li: doi:10.4169/074683410X488683 arxiv:0903.0753
ab: Summary: Viviani’s theorem states that the sum of distances from any
point inside an equilateral triangle to its sides is constant. Here, in
an extension of this result, the author shows, using linear
programming, that any convex polygon can be divided into parallel line
segments on which the sum of the distances to the sides of the polygon
is constant. A polygon has the CVS property if the sum of distances
from any inner point to its sides is constant. An amazing converse of
Viviani’s theorem is deduced: if just three non-collinear points
inside a convex polygon have equal sums of distances then the polygon
has the CVS property. For concave polygons the situation is quite
different. For polyhedra analogous results are deduced.
rv: