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在《中等数学》1983年第2期《勾股定理的新探索》一文的基础上,我们来研究余弦定理在三维空间的推广。首先,观察一个三角形,它有不共线的三个顶点,每个顶点对应着三角形的一条边,每两边又相交成三角形的一个角。其次,比较一个四面体,它有不共面的四个顶点,每个顶点对应着四面体的一个面,每两个面又相交成一个二面角。再次,余弦定理是考虑三角形边长与夹角之间的关系式,在三维空间中,则应考虑四面体的面的面积和夹角之间的关系式。
On the basis of the article “The New Exploration of the Pythagorean Theorem” in the second issue of “Middle Mathematics” (1983), we will study the spread of the cosine theorem in three-dimensional space. First, observe a triangle that has three vertices that are not collinear. Each vertex corresponds to a side of a triangle, and each side intersects with a corner of a triangle. Next, compare a tetrahedron, which has four vertices that are not coplanar. Each vertex corresponds to a face of a tetrahedron, and each face is also intersected by a dihedral angle. Again, the cosine theorem considers the relationship between the length of the sides of the triangle and the included angle. In a three-dimensional space, the relationship between the area and the included angle of the tetrahedron should be considered.