Problem 5

Let

a,b,c

be the side lengths of a triangle. Prove that

\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 5 of 5: Determine equality

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a+b-c=b+c-a=c+a-b\Longleftrightarrow a=b=c

Detailed analysis

Equality in the sum requires equality in all three applications of the lemma, so $a+b-c=b+c-a=c+a-b$ . Subtracting these equalities gives $a=b=c$ . Conversely, an equilateral triangle makes all three lemma equalities hold, so equality is attained exactly when $a=b=c$ .