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.
Deep Dive into On Vivianis Theorem and its Extensions.
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.
Let P be a polygon or polyhedron, consisting of both boundary and interior points. Define a distance sum function V : P → R, where for each point P ∈ P the value V(P ) is defined as the sum of the distances from the point P to the sides (faces) of P.
We say that P has the constant Viviani sum property, abbreviated by the “CVS property”, if and only if the function V is constant.
Viviani (1622-1703), who was a student and assistant of Galileo, discovered the theorem which states that equilateral triangles have the CVS property. The theorem can be easily proved by an area argument; Joining a point P inside the triangle to its vertices divides it into three parts, the sum of their areas will be equal to the area of the original one. Therefore, V(P ) will be equal to the height of the triangle and the theorem follows. The importance of Viviani’s theorem may be derived from the fact that his teacher Torricelli (1608-1647) used it to locate the Fermat point of a triangle [2, pp. 443].
Samelson [6, pp. 225] gave a proof of Viviaini’s theorem that uses vectors and Chen & Liang [1, pp. 390-391] used this vector method for proving the converse of the theorem; If inside a triangle there is a circular region in which V is constant then the triangle is equilateral.
Kawasaki [3, pp. 213], by a proof without words, uses only rotations to establish Viviani’s theorem. There is an extension of the theorem to all regular polygons, by the area method: All regular polygons have the CVS property. There is also an extension of the theorem to regular polyhedrons, by a volume argument: All regular polyhedra have the CVS property. Kawasaki, Yagi and Yanagawa [4, pp. 283] gave a different proof for the regular tetrahedron.
What happens for general polygons and polyhedra? Surprisingly, there is a strict correlation between Viviani’s theorem, its converse and extensions to linear programming.
This correlation is manifested by the following main result:
Theorem 1.1 (a) Any convex polygon can be divided into parallel segments such that V is constant on each segment. (b) Any convex polyhedron can be divided into parallel cross sections such that V is constant on each cross section.
These segments or cross sections, on which V is constant, will be called isosum layers ( or definitely isosum segments and isosum cross sections). They are formed by the intersection of P and a suitable family of parallel lines (planes). The value of the function V will increase when passing, in some direction, from one isosum layer into another, unless P has the CVS property.
The correlation soon will be clear. Each linear programming problem is composed of an objective function and a feasible region (see for example [5] or [7]). Moreover, the objective function divides the feasible region into isoprofit layers, these layers are parallel and consist of points on which the objective function has constant value. Furthermore, moving in some direction will increase the value of the objective function unless it is constant in the feasible region.
Because of this correlation we thus conclude the following amazing result, concerning the converse of Viviani’s theorem. The theorem tells us that measuring the distances from the sides of three non-collinear points, inside a convex polygon, is sufficient for determining if the polygon has the CVS property. Likewise, measuring the distances from the faces of four non-coplanar points, inside a convex polyhedron, is sufficient for determining if the polyhedron possesses the CVS property.
We then end with the following beautiful conclusions. (c) If a convex polygon possesses a reflection symmetry across an axis l, then the polygon has the CVS property or otherwise the isosum segments are perpendicular to l.
While for polyhedra we have, Corollary 1.4 (a) If there is an isometry of the space which fixes the polyhedron but not an isosum cross section, then the polyhedron has the CVS property.
(b) If a convex polyhedron possesses two rotational symmetries, around different axes, then the polyhedron has the CVS property.
From Corollary 1.3, one can deduce that all regular polygons have the CVS property. Besides, any parallelogram has this property, since it possesses a rotational symmetry around its centroid by an angle of 180 • .
Obviously, the existence of two reflection symmetries, by different axes, of a polygon will imply a rotational symmetry and hence the polygon must own the CVS property.
Moreover, for triangles and quadrilaterals, the existence of a rotational symmetry characterizes the possessing of CVS property, since in these cases the polygons would be only equilateral triangles and parallelograms.
Consequently, an n-gon, for n ≥ 5, that does not possess the CVS property must have at most one symmetry which is the reflection symmetry.
Analogously, by Corollary 1.4, all regular polyhedra and regular prisms have the CVS property. Likewise, any parallelepiped has the CVS property. Since it possesses thr
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