Let and be lengths of the sides of a triangle. Suppose that . Show that
I was thinking something like this. Since each side isn't really defined to what it could be, and each side wouldn't exactly be the same length. We could say that C being the longest side will have a length of .25. C gets used twice, so you have .5. A will be .2, used it twice you get .4. Now you have a total of .9. Which leaves B to the shortest length getting .05, used twice will get you .1. With a total of 1. So now if we just plugin what we substituted earlier.
(.2+1)(.05+1)(.25+1) < 4
(1.2) (1.05) = 1.26 (1.25) = 1.575 < 4?
Though I'm not completely sure this is how it should be done, and if I made a mistake anywhere please feel free to correct me. This is how I would go about doing it though.
I would rewrite:
With the given (2)
Substituting (2) in (1) gives:
This looks very hard like if you write:
But I can't see any relation ...
But now I did nothing wit the fact a,b,c are the sides of a triangle. I don't know of this make sense, it was just an attempt
Do you have a hint or something? Or is the problem also not clear to you? ...
We may assume without loss of generality that ; from , we see then that .
Sppose . Since are the side lengths of a triangle, we have and then ; therefore . From , it follows that are all less than .
Now if , then clearly . This together with the above means that the product is negative.
Write ; similarly, expand . We have .
But which implies that .
As melese has pointed out, the key thing about a triangle is that each side must be shorter than the sum of the other two sides. So . Write that as , and let . Then , and similarly , Therefore
. . . . . . . . . .
Thus , and so
However, if x>0 then It follows that Thus and so As shown in Siron's comment above, that is equivalent to
Incidentally, I think it's more realistic to allow equalities in your assumption. That is, you really should assume