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Volume 10 (2009), Issue 1, Article 16, 6 pp.AN EQUIVALENT FORM OF THE FUNDAMENTAL TRIANGLE INEQUALITY
AND ITS APPLICATIONS
SHAN-HE WU AND MIHÁLY BENCZE
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE
L ONGYAN U NIVERSITY
L ONGYAN F UJIAN 364012
P EOPLE ’ S R EPUBLIC OF C HINA
wushanhe@yahoo.com.cn
URL: http://www.hindawi.com/10865893.html
S TR . H ARMANULUI 6
505600 S ACELE -N ÉGYFALU
J UD . B RASOV, ROMANIA
benczemihaly@yahoo.com
Received 12 March, 2008; accepted 20 January, 2009
Communicated by S.S. Dragomir
A BSTRACT. An equivalent form of the fundamental triangle inequality is given. The result is
then used to obtain an improvement of the Leuenberger’s inequality and a new proof of the
Garfunkel-Bankoff inequality.
Key words and phrases: Fundamental triangle inequality, Equivalent form, Garfunkel-Bankoff inequality, Leuenberger’s in-
equality.
2000 Mathematics Subject Classification. 26D05, 26D15, 51M16.
1. I NTRODUCTION AND M AIN R ESULTS
In what follows, we denote by A, B, C the angles of triangle ABC, let a, b, c denote the
lengths of its corresponding sides, and let s, R and r denote respectively the semi-perimeter,
circumradius and inradius of a triangle. We will customarily use the symbol of cyclic sums:
X X
f (a) = f (a) + f (b) + f (c), f (a, b) = f (a, b) + f (b, c) + f (c, a).
The fundamental triangle inequality is one of the cornerstones of geometric inequalities for
triangles. It reads as follows:
√
(1.1) 2R2 + 10Rr − r2 − 2(R − 2r) R2 − 2Rr
√
6 s2 6 2R2 + 10Rr − r2 + 2(R − 2r) R2 − 2Rr.
The present investigation was supported, in part, by the Natural Science Foundation of Fujian Province of China under Grant S0850023,
and, in part, by the Science Foundation of Project of Fujian Province Education Department of China under Grant JA08231.
Both of the authors are grateful to the referees for their helpful and constructive comments which enhanced this paper.
275-08
, 2 S HAN -H E W U AND M IHÁLY B ENCZE
The equality holds in the left (or right) inequality of (1.1) if and only if the triangle is isosce-
les.
As is well known, the inequality (1.1) is a necessary and sufficient condition for the existence
of a triangle with elements R, r and s. This classical inequality has many important applica-
tions in the theory of geometric inequalities and has received much attention. There exist a
large number of papers that have been written about applying the inequality (1.1) to establish
and prove the geometric inequalities for triangles, e.g., see [1] to [10] and the references cited
therein.
In a recent paper [11], we presented a sharpened version of the fundamental triangle, as
follows:
√
(1.2) 2R2 + 10Rr − r2 − 2(R − 2r) R2 − 2Rr cos φ
√
6 s2 6 2R2 + 10Rr − r2 + 2(R − 2r) R2 − 2Rr cos φ,
where φ = min{|A − B| , |B − C| , |C − A|}.
The objective of this paper is to give an equivalent form of the fundamental triangle inequal-
ity. As applications, we shall apply our results to a new proof of the Garfunkel-Bankoff inequal-
ity and an improvement of the Leuenberger inequality. It will be shown that our new inequality
can efficaciously reduce the computational complexity in the proof of certain inequalities for
triangles. We state the main result in the following theorem:
Theorem 1.1. For any triangle ABC the following inequalities hold true:
1 s2 1
(1.3) δ(4 − δ)3 6 2 6 (2 − δ)(2 + δ)3 ,
4 R 4
p
where δ = 1 − 1 − (2r/R). Furthermore, the equality holds in the left (or right) inequality
of (1.3) if and only if the triangle is isosceles.
2. P ROOF OF T HEOREM 1.1
We rewrite the fundamental triangle inequality (1.1) as:
r
r2
10r 2r 2r
(2.1) 2+ − 2 −2 1− 1−
R R R R
r
s2 r2
10r 2r 2r
6 2 62+ − 2 +2 1− 1− .
R R R R R
By the Euler’s inequality R > 2r (see [1]), we observe that
2r
061− < 1.
R
Let
r
2r
(2.2) 1− = 1 − δ, 0 < δ 6 1.
R
Also, the identity (2.2) is equivalent to the following identity:
r 1
(2.3) = δ − δ2.
R 2
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 16, 6 pp. http://jipam.vu.edu.au/
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