Page 45 - RMGO 5

Some identities and inequalities in triangle 45

Theorem 3. In any triangle ABC hold the inequalities:

3 X a X a + b
1. + ≥ 2 ;
2 b + c a + b + 2c

X a X a + b
2. 3 + ≥ ;
b + c − a c
a + b + c X a 2 X (a + b) 2
3. + ≥ ;
2 b + c a + b + 2c

a 2 1 (a + b) 2
X X
4. a + b + c + ≥ .
−a + b + c 2 c
x x x 2
Proof. The functions f(x) = (for 1.), g(x) = (for 2.), h(x) =
k − x k − 2x k − x
x 2
(for 3.), m(x) = (for 4.) where k = a + b + c are convex, and we apply the
k − 2x
Popoviciu’s Inequality.

References

[1] Octogon Mathematical Magazine (1993-2021).