If $a,\,b$ and $c$ are the sides of a triangle $ABC$, prove that if $a^2+b^2>5c^2$, then $c$ is the length of the shortest side.
Suppose we assume:
\(\displaystyle a>c\implies a^2>c^2\)
\(\displaystyle b>c\implies b^2>c^2\)
These two conditions also imply:
Adding the three implications, we obtain:
The triangle inequality implies:
And, we are given:
Adding these last two, there results:
As we initially found, this is the result of assuming both \(a\) and \(b\) are greater than \(c\). :)
You have assumed c being the shortest side and you have taken the given condition as well. So I am not convinced that
the proof is right. You have taken both the condition and assumption and proved it.
If I have missed something kindly let me know.
I have assumed that \(c\) is the shortest side and shown how it leads to an implication provided both by the given and the triangle inequality. It seems to me this is sufficient. Is it not?
But does not prove that this is not true if c is not the shortest side which we need to prove