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将四面体的每一组对棱之间的距离(即公垂线的长度)叫做四面体的一个“宽度”。本文主要由一些引理得到了关于四面体“宽度”的两个不等式。命题一设四面体ABCD的三个宽度为d_1,d_2,d_3,体积为V,则有 d_1d_2d_3≤3V, (1)当且仅当四面体的各对对棱相等时,等号成立。为证命题,先看如下两个引理。引理1 若四面体的体积为v,其一组对棱之长分别为a,b,此组对棱间的距离为d,夹角为a,则有 V=1/6abdsina, (2) 引理 2设四面体体积为V,六条棱长的乘积为P,三对对棱成角分
The distance between each set of pairs of tetrahedrons (i.e., the length of a vertical line) is called a “width” of the tetrahedron. In this paper, we obtain two inequalities about the “width” of the tetrahedron from some lemmas. The proposition one sets the three widths of the tetrahedron ABCD to d_1, d_2, d_3, and the volume to V. Then there is d_1d_2d_3≤3V. (1) If and only if the tetrahedron pairs are equal, the equal sign holds. To prove the proposition, look at the following two lemmas. Lemma 1 If the volume of the tetrahedron is v, and the length of its pair of edges is a, b, the distance between the pair of edges is d, and the angle is a, then there is V = 1/6abdsina, (2) Lemma 2 sets the tetrahedral volume to V, the product of six edges long to P, and the three pairs of angled angles