Page 12 - MATINF Nr. 8

12 M. Bencze

PB
= cos ω − sin ωctg C,
a
AP
= cos ω − sin ωctg B.
c
Adding these we get

AP PB CP X
+ + = 3 cos ω − sin ω ctg A = 2 cos ω.
c a b
□

Corollary 2. In any triangle ABC, we have
27 · AP · BP · CP
3
cos ω ≥ .
8abc
Proof. By AM-GM Inequality,
…
AP AP
X 3 Y
2 cos ω = ≥ 3 .
c c
□

Corollary 3. In any triangle ABC, we have

2
2 2 2
4a b c cos ω
2
2
2
AP + BP + CP ≥ P .
a b
2 2
Proof. By Cauchy-Schwarz Inequality,
Å ã
AP 1
X ÄX ä X
2 cos ω = ≤ AP 2 .
c a 2
□
Corollary 4. In all triangle ABC, we have
√ √ √ »
AP + BP + CP ≤ 2(a + b + c) cos ω.

Proof. By Cauchy-Schwarz Inequality,
P √
X AP ( AP) 2
2 cos ω = ≥ P .
c c

□
Corollary 5. In all triangle ABC, we have

a
c
P
Å ã a Å ã Å ã Å ã b
2 cos ω AP BP CP
≥ .
P
a c 2 a 2 b 2
Proof. By weighted AM-GM Inequality, we get
c
P AP Å Å ã ã 1
2 cos ω c · c 2 Y AP c
P
= ≥ .
P P
a c c 2
□