MathLabs

Problem 6

Let a,b,ca,b,c be the side lengths of a triangle. Prove that a2b(a−b)+b2c(b−c)+c2a(c−a)≥0a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0. Determine when equality occurs.
Step 3 of 5: Prepare Cauchy-Schwarz
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xy3+yz3+zx3=(yxyz)2z+(zxyz)2x+(xxyz)2yxy^3+yz^3+zx^3=\frac{(y\sqrt{xyz})^2}{z}+\frac{(z\sqrt{xyz})^2}{x}+\frac{(x\sqrt{xyz})^2}{y}
Detailed analysis

Rewrite each term as a square divided by a positive variable. The denominators add to x+y+zx+y+z, while the numerators are chosen so that their sum is xyz(x+y+z)\sqrt{xyz}(x+y+z).