Page 30 - MATINF Nr. 8

30 5 FINAL REMARKS

As for the lower bound, we have that

 Ä √ ä
 − sin α cos 2 π + α , if 0 < α < arccot 3 3
4

inf [cos (A + α) cos (B + α) cos (C + α)] =
Ä √ ä
 π π
cos + α , if
 3 arccot 3 3 < α ≤
3 3
over all non-obtuse-angled triangles ABC.
Finally, the following is true over all triangles ABC :

3
inf [cos (A + α) cos (B + α) cos (C + α)] = − cos α.

References

[1] L. Giugiuc, C. A. Tr˘anc˘an˘au, A. Pˆırvuceanu, Metode moderne de demonstrare a unor
inegalit˘at , i in triunghi, Revista de Matematic˘a Marinescu-Ghemeci Octavian (2018), nr. 1,
41–43.

[2] C. Mateescu, Useful Identities and Inequalities in Geometry Thread, https://
artofproblemsolving.com/community/c6h412623p2481722

[3] C. Mateescu, Useful Identities and Inequalities in Geometry Thread, https://
artofproblemsolving.com/community/c6h412623p2382562

[4] D. S. Mitrinovi´c, E. Peˇcari´c, V. Volenec, Recent Advances in Geometric Inequalities, Kluwer
Academic Publishers, 1989.