110. In acute-angled triangle ABC the following relationship holds
X
2
4R + r
1
√
6 (ma + wb + hc ) ·
.
3sr2
s−a
Solution. Let ra , rb , rc and F denote the exradii and the area of the triangle
respectively. We use the well-known inequality
ha + hb + hc > 9r,
and the equality
ra + rb + rc = 4R + r.
Hence by the Cauchy-Schwarz inequality
4R + r
4R + r
> (ha + hb + hc ) ·
(ma + wb + hc ) ·
2
3sr
3sr2
4R + r
ra + rb + rc
> 3·
=3·
F
F
X
2
X 1
1
√
= 3·
>
,
s−a
s−a
and we are done.
Martin Lukarevski, Department of Mathematics and Statistics, University ”Goce
Delcev” - Stip, Macedonia
E-mail address: [email protected]
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