If a, b, c are the sides of a triangle, and (a+b+c)^3 ≥λ(a+b-c)(b+c-a)(c+a-b), then λequals (a)

step 1 Hi, we're given here APC outside of a triangle. We're given this inequality here and solve for L

If a, b, c are the sides of a triangle, and (a+b+c)^3...

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of a triangle, and $(\mathrm{a}+\mathrm{b}+\mathrm{c})^{3} \geq \lambda(\mathrm{a}+\mathrm{b}-\mathrm{c})(\mathrm{b}+\mathrm{c}-\mathrm{a})(\mathrm{c}+\mathrm{a}-\mathrm{b})$, then $\lambda$ equals
(a) 9
(b) 3
(c) 8
(d) 27

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Key Concepts

Symmetric Inequalities

Symmetric inequalities are those expressions that remain invariant under any permutation of the involved variables. In the realm of triangle side lengths, such expressions often simplify when all sides are equal, which provides a powerful method for determining whether an inequality holds in general, and for identifying the best constant for the inequality.

Optimization through Extremal Cases

Optimization in inequalities often involves testing the boundaries or extremal cases of the variables involved. In geometric problems related to triangle sides, considering an equilateral triangle—a case where all sides are equal and symmetry is at its maximum—can reveal the critical values or constants which optimize the inequality, ensuring that it holds for all valid triangles.

Triangle Inequality

The triangle inequality is a core principle in geometry stating that the sum of any two side lengths of a triangle must be greater than the third side. This condition is essential to ensure that three given segments can form a triangle and is a foundational constraint in many geometric problems and inequalities involving triangle sides.

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