MathLabs

Problem 5

Let a,b,ca,b,c be the side lengths of a triangle. Prove that a+b−c+b+c−a+c+a−b≤a+b+c\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}\le\sqrt a+\sqrt b+\sqrt c, and determine when equality occurs.
Step 4 of 5: Sum the three estimates
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a+b−c+b+c−a+c+a−b≤a+b+c\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}\le\sqrt a+\sqrt b+\sqrt c
Detailed analysis

Adding the three inequalities gives exactly the required inequality, because the three radicands occur twice with coefficient 1/21/2 on the right-hand sides.