Page 25 - MATINF Nr. 9-10

New inequalities in triangle 25

2
2
2

1 − cos C ≤ (1 + cos C) cos A + cos B , and hence
2
sin C
2
2
≤ cos A + cos B.
1 + cos C

Corollary 5. In any non-obtuse triangle ABC hold the following inequalities:
2
2
2
2
2
s + r − 4R 2 X sin A 2R − s + (2R + r) 2
≤ ≤
2R 2 1 + cos A R 2
2
2
2
(a refinement of the inequality 3s ≤ 16R + 8Rr + r ). The first inequality is valid in any
triangle.
Theorem 3. In any acute triangle ABC hold the following inequalities:
2
2
sin A sin A
≤ 1 + cos A ≤ .
2
cos B + cos C 2 cos B cos C
2
Proof. The inequalities follow from Theorem 2.
Corollary 6. In any acute triangle ABC hold the following inequalities:
2
2
sin A r 1 sin A
X X
≤ 4 + ≤ .
2
2
cos B + cos C R 2 cos B cos C
Theorem 4. In any acute triangle ABC holds the following inequality:
Å ã
cos A cos B cos C R
max , , ≤ p .
2
2
2
2
2
2
1 + sin B + sin C 1 + cos C + cos A 1 + cos A + cos B 2 3r(2R − r)
2
s − r(4R + r)
P 2
Proof. We have the identity sin A = .
2R 2
2
2
By Gerretsen’s inequality s ≥ 16Rr − 5r we derive that
3r(2R − r)
X
2
sin A ≥ .
R 2
It follows that
3r(2R − r)
2
2
2
2
2
cos A + sin A + sin B + sin C ≥ + cos A, so
R 2
3r(2R − r) 2 »
2
2
2
1 + sin B + sin C ≥ + cos A ≥ 3r(2R − r) · cos A, and hence
R 2 R
cos A R
.
2
2
≤ p
1 + sin B + sin C 2 3r(2R − r)

Corollary 7. In any acute triangle ABC hold the following inequalities:
2
2
2
1 R (s + v − 4R )
P
1) î ó;
2
2
1 + sin B + sin C ≤ p 2 2
2 3r(2R − r) s − (2R + r)
cos A 3R
P
2) ≤ p ;
2
2
1 + sin B + sin C 2 3r(2R − r)
2
cos A R + r
P
3) ≤ p .
2
2
1 + sin B + sin C 2 3r(2R − r)