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 2 of 6: Rewrite the Heron product
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16S2=4b2c2−(b2+c2−a2)216S^2=4b^2c^2-(b^2+c^2-a^2)^2
Detailed analysis

Pair the factors as ((b+c)2−a2)(a2−(b−c)2)((b+c)^2-a^2)(a^2-(b-c)^2) and use the difference-of-squares identity. This gives 16S2=4b2c2−(b2+c2−a2)216S^2=4b^2c^2-(b^2+c^2-a^2)^2.