MathLabs

Problem 2

Let a,b,ca,b,c be the side lengths of a triangle whose area is SS. Prove that a2+b2+c2≥4S3a^2+b^2+c^2\ge4S\sqrt{3}. In what case does equality hold?
Step 1 of 6: Apply Heron's formula
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In plain words

Heron's formula turns the area into an expression involving only the side lengths.

16S2=(a+b+c)(−a+b+c)(a−b+c)(a+b−c)16S^2=(a+b+c)(-a+b+c)(a-b+c)(a+b-c)
Detailed analysis

Heron's formula gives S2=s(s−a)(s−b)(s−c)S^2=s(s-a)(s-b)(s-c) with s=(a+b+c)/2s=(a+b+c)/2. Multiplying the four factors by 22 yields the displayed identity.