Page 23 - MATINF Nr. 9-10

New inequalities in triangle

Mih´aly Bencze 1

In this paper we present some new inequalities in triangle.

Theorem 1. In any triangle ABC hold the following inequalities:

ß ™
sin A sin B sin C
max , ,
2
2
2
2
2
2
1 + cos B + cos C 1 + cos C + cos A 1 + cos A + cos B
R 1
≤ √ ≤ √ .
2 R − r 2 3
2
2
s − (2R + r) 2
P 2
Proof. We have the identity cos A = 1 − .
2R 2
2
2
2
By Gerretsen’s inequality s ≤ 4R + 4Rr + 3r we derive that
2
X r
2
cos A ≥ 1 − .
R
It follows that
2
R − r 2
2
2
2
2
2
sin A + cos A + cos B + cos C ≥ + sin A, so
R 2 √
2
2
R − r 2 2 R − r 2
2
2
2
1 + cos B + cos C ≥ + sin A ≥ · sin A, and hence
R 2 R
sin A R 1
≤ √ ≤ √ ,
2
2
1 + cos B + cos C 2 R − r 2 3
2
where the last inequality holds from Euler’s R ≥ 2r inequality.
Corollary 1. In any triangle ABC hold the following inequalities:
sin A 3R √
P
1) ≤ √ ≤ 3
2
2
1 + cos B + cos C 2 R − r 2
2
(a refinement of the inequality from problem O.600 from Mathematical Reflections);
2
sin A s
P
2) ≤ √ ;
1 + cos B + cos C 2 R − r 2
2
2
2
A
sin A sin 2
P 2 2R − r
3) ≤ √ ;
2
2
1 + cos A + cos C 4 R − r 2
2
A
sin A cos 2
P 2 4R + r
4) ≤ √ .
2
2
2
1 + cos B + cos C 4 R − r 2
1
Profesor dr., Bras , ov, benczemihaly@gmail.com
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