MathLabs

Problem 2

Suppose a,b,ca,b,c are the sides of a triangle. Prove that a2(b+c−a)+b2(c+a−b)+c2(a+b−c)≤3abca^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le3abc.
Step 2 of 4: Rewrite the inequality
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2z(x+y)2+2y(x+z)2+2x(y+z)2≤3(x+y)(x+z)(y+z)2z(x+y)^2+2y(x+z)^2+2x(y+z)^2\le3(x+y)(x+z)(y+z)
Detailed analysis

Substitution gives exactly the displayed inequality, because b+c−a=2zb+c-a=2z, c+a−b=2yc+a-b=2y, and a+b−c=2xa+b-c=2x.