Problem 6

Let

a,b,c

be the side lengths of a triangle. Prove that

a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0

. Determine when equality occurs.

Step 5 of 5: Equality case

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\frac{y\sqrt{xyz}}{z}=\frac{z\sqrt{xyz}}{x}=\frac{x\sqrt{xyz}}{y}\Longleftrightarrow x=y=z\Longleftrightarrow a=b=c

Detailed analysis

Equality in Cauchy-Schwarz requires $u_1/v_1=u_2/v_2=u_3/v_3$ , namely $y/z=z/x=x/y$ . Since $x,y,z>0$ , this is equivalent to $x=y=z$ , which gives $a=b=c$ . Conversely an equilateral triangle makes every difference vanish, so equality occurs exactly for equilateral triangles.