Page 21 - MATINF Nr.2

Dezvolt˘ari din RMM, Winter 2017 21

Solut , ie: Avem

x Ç x å X x + y + z
X 2 2 X 2 2 2 2 X 2 2
a (b + c) = + 1 − 1 a (b + c) = a (b + c) − a (b + c)
y + z y + z y + z

Bergstrom ( P a (b + c)) 2 X 2
2
≥ (x + y + z) − a (b + c)
(y + z)
P
2
2
(2 (p + r + 4Rr)) 2 î 2 ó
4
2 2
2
= (x + y + z) − 2 p + 2p r + r (4R + r)
2 (x + y + z)
1 Ä Ä ää 2 î ó
2
2
2
2 2
2
2
4
2 2
= 2 p + r + 4Rr − 2 p + 2p r + r (4R + r) 2 = 16p Rr ≥ 32p r = 32S .
2
Mai sus am folosit identit˘at , ile cunoscute ˆın triunghi:
a (b + c) = 2 bc = 2 (p + r + 4Rr) s , i a (b + c) = 2 p + 2p r + r (4R + r) .
P P 2 2 P 2 2 î 4 2 2 2 2 ó
Egalitatea are loc dac˘a s , i numai dac˘a triunghiul este echilateral.
Bibliografie
[1] D.M. B˘atinet , u-Giurgiu, M. Lukarevski, Romanian Mathematical Magazine, Founding Editor
Daniel Sitaru, JP. 105, Winter Edition 2017.

[2] O. Bottema, R.Z. Djordjevic, R.R. Janic, D.S. Mitrinovic, P.M. Vasic, Geometric Inequalities,
The Netherlands, Groningen, 1969.

[3] M. Chirciu, Inegalit˘at , i algebrice, de la init , iere la performant , ˘a, Editura Paralela 45, Pites , ti,
2014.

[4] M. Chirciu, Inegalit˘at , i cu laturi s , i raze ˆın triunghi, de la init , iere la performant , ˘a, Editura
Paralela 45, Pites , ti, 2017.

[5] G. Tsintsifas, Crux Mathematicorum, Nr. 11/1986.