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.
-Auri
I would rewrite:
as:
(1)
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? ...
Here's what I've tried.
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
However, it's still not clear to me how you get that all three have to be less than one. Let's say you assume a < b < c. Then you can easily show that a and b must be less than one. Why does c have to be less than one? And if you assume a different ordering, well, that's a separate case. In each possible ordering, you still get one side that could, theoretically, be greater than one. The fact that they're disjoint cases is neither here nor there.
Incidentally, I think it's more realistic to allow equalities in your assumption. That is, you really should assume