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The existence of triangles, tetrahedra, and higher-dimensional simplices with prescribed exradii

Published online by Cambridge University Press:  18 June 2018

Mowaffaq Hajja
Mowaffaq Hajja*
Affiliation:
Department of Basic Sciences and Mathematics, Philadelphia University, P. O. Box 1, 19392, Amman, Jordan e-mail: mowhajja@yahoo.com
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Extract

This Article is inspired by a problem that appeared recently in the American Mathematical Monthly, namely 11972 in [1, p. 369]. The problem asks the readers to prove that if r is the inradius of a tetrahedron, and if r1, r2, r3, r4 are its exradii, then

By taking r = r1 = r2 = r3 = r4 = 1, one sees that (1) is not true for all positive numbers r, r1, r2, r3, r4. This is not surprising, since r is dependent on r1, r2, r3, r4 by the elegant relation

which we shall prove in Theorem 2 below; see also, for example, [2, (5), §266, p. 92], [3. §41, 2°, (1), p. 76] and [4, Problem 6′(i), p. 39]. Using this relation, one can rewrite (1) as

Intuitively, the four numbers r1, r2, r3, r4 are independent, and one may thus ask whether the inequality (3) holds for all positive numbers r1, r2, r3, r4 regardless of being the exradii of some tetrahedron or not.

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 554 , July 2018 , pp. 257 - 263
Copyright
Copyright © Mathematical Association 2018 

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References

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