关于Viviani定理及其推广
On Viviani's Theorem and its Extensions
摘要：
维维亚尼定理指出，从一个等边三角形内的任何一点到它的边的距离之和是常数。我们考虑定理的扩展，并表明任何凸多边形可以分为平行的部分，使每段上的点的距离的总和是恒定的。一个多边形具有CVS属性，如果从任何内部点到它的边的距离之和是常数。导出了Viviani定理的逆定理的一个令人惊奇的结果：凸多边形内到各边的距离之和相等的三个非共线点足以具有CVS性质。对于凹多边形，情况则完全不同，而对于多面体，则推导出类似的结果。
英文摘要：
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.
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