| Inequalities for inscribed simplexes |
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| 作者姓名： | YANGShiguo |
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| 作者单位： | DepartmentofMathematics,AnhuiInstituteofEducation,Hefei230061,P.R.China |
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| 摘 要： | The problem on the geometrc inequalities involving an n-dimensional simplex and its inscribed simplex is studied. Aninequality is established, which reveals that the difference between the squared circumradius of the n-dimensional simplex andthe squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not lessthan the 2(n-1)th power of n times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is atangent point one. Deduction from this inequality reaches a generalization of n-dimensional Euler inequality indicating that thecircumradius of the simplex is not less than the n-fold inradius. Another inequality is derived to present the relationship betweenthe circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex.
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| 关 键 词： | 内接单纯形 内径 外接圆半径 几何不等式 切点 |
Inequalities for inscribed simplexes |
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| YANGShiguo.Inequalities for inscribed simplexes[J].Journal of Chongqing University,2004,3(1):86-88. |
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| Authors: | Yang Shiguo |
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| Abstract: | The problem on the geometrc inequalities involving ann-dimensional simplex and its inscribed simplex is studied. An inequality is established, which reveals that the difference between the squared circumradius of then-dimensional simplex and the squared distance between its circumcenter and barycenter times the squared circumradius of its inscribed simplex is not less than the 2(n-1)th power ofnn times its squared inradius, and is equal to when the simplex is regular and its inscribed siplex is a tangent point one. Deduction from this inequality reaches a generalization ofn-dimensional Euler inequality indicating that the circumradius of the simplex is not less than then-fold inradius. Another inequality is derived to present the relationship between the circumradius of the n-dimensional simplex and the circumradius and inradius of its pedal simplex. |
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| Keywords: | simplex inscribed simplex inradius circumradius inequality derived present relationship pedal simplex Deduction generalization Euler inequality regular tangent point power inradius difference distance times circumradius established problem inequalities involving |
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