平面几何中,有一个欧拉不等式: 设△ABC的外接圆和内切圆的半径分别是R和r,则 R≥2r。其中等号当且仅当△ABC是正三角形时成立。这个结论在三维空间中可推广如下: 设四面体A_1—A_2A_3A_4(简记四面体A,下同)的外接球和内切球的半径分别是R和r,则 In plane geometry, there is an Euler inequality: Let R and r be the radius of the circumscribed circle and inscribed circle of △ABC, then R ≥ 2r. The equal sign holds if and only if ΔABC is an equilateral triangle. This conclusion can be generalized in three-dimensional space as follows: Let the radius of the circumcircle and the inscribed sphere of the tetrahedron A_1-A_2A_3A_4 (abbreviation of tetrahedron A, the same below) be R and r, respectively.