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Some identities and inequalities in triangle

Mih´aly BENCZE 1

In this paper we present some new inequalities in triangle. We starting with a
well-known problem.

Problem (M 7, Kvant). In any triangle ABC holds the inequality

X a
≥ 3.
b + c − a
Solution. If 2a = y + z, 2b = z + x, 2c = x + y then

X a 1 X y + z 1 X x y 1 X
= = + ≥ 2 = 3.
b + c − a 2 x 2 y x 2
In this paper we give another proof and another relations, inequalities and
generalizations.

Theorem 1. In any triangle ABC we have the relations:

X 1 4R + r
1. = ;
s − a sr
X 1 1
2. = ;
(s − a)(s − b) r 2
2
X 1 (4R + r) − 2s 2
3. = ;
2 2
(s − a) 2 s r
2
2
1 s − 2r − 8Rr
X
4. = ;
2 4
2
(s − a) (s − b) 2 s r
X a 2(2R − r)
5. = ;
s − a r
Y a 4R
6. = ;
s − a r
a(s − a) 2(2R − r)
X
7. = ;
(s − b)(s − c) r
1
Profesor dr., Bras , ov, benczemihaly@gmail.com
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