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 1 of 5: Apply Ravi substitution
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a=y+z,b=z+x,c=x+y,x,y,z>0a=y+z,\qquad b=z+x,\qquad c=x+y,\qquad x,y,z>0
Detailed analysis

The strict triangle inequalities are equivalent to the existence of positive x,y,zx,y,z with a=y+z,b=z+x,c=x+ya=y+z,b=z+x,c=x+y. This parametrizes every nondegenerate triangle without loss.