MHB Triangle Inequality Proof: Shortest Side in Relation to Sides a, b, and c

[anemone]

Here is this week's POTW:

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The sides $a,\,b$ and $c$ of a triangle satisfy $a^2+ b^2> 5c^2$. Prove that $c$ is the shortest side of this triangle.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to kaliprasad for his correct solution!:)

Here's the model solution provided by my best friend, Michelle:

Spoiler

It is clear that $c$ cannot be the longest side. So if $c$ is not the shortest, WLOG, we can assume that $a > c > b$.
Hence $(b+c)^2\le 2b^2+ 2c^2< 4c^2< 4c^2 + (c^2 - b^2) < a^2$.

Therefore we get $b+c < a$ which is a contradiction for $a,\,b$, and $c$ are sides of a triangle.

We can conclude by now that $c$ is the shortest side of that triangle.