Show that if in an acute triangle then the area of the triangle is greater than 1 ( ). (I'm not sure if this involves trigonometry at all.)

(I know that because ).

Printable View

- October 27th 2009, 01:21 AMjames_bondProve that if abc=a+b+c in an acute triangle then 1<S
Show that if in an acute triangle then the area of the triangle is greater than 1 ( ). (I'm not sure if this involves trigonometry at all.)

(I know that because ). - October 27th 2009, 02:58 AMproscientia
What is

- October 27th 2009, 04:35 AMjames_bond
The area of the triangle.

- October 27th 2009, 04:55 AMproscientia
By Heron’s formula for the area of the triangle,

By AM–GM,

where

But, again by AM–GM,

Now, Thus if you can show that you are done. I think this should be straightforward. - October 27th 2009, 01:53 PMjames_bond
Sorry for not noticing sooner but your solution appears to be incorrect. By Heron's formula:

which implies which is obviously false (for a general ).

I'm eagerly waiting for a good solution. (The upper was so simple to get I assume the lower should come just as easy but I can't see :( ) Any help would be greatly appreciated! Thanks. - October 28th 2009, 10:14 AMproscientia
Yes, I’ve just noticed that. It was very careless of me. Sorry.

- October 29th 2009, 04:24 AMproscientia
Hmm, I wondering if what is to be proved is actually false. Heron’s formula gives (hopefully that’s correct this time). Then However it is possible that this inequality may not be true for arbitrarily close to See this.

- October 29th 2009, 09:48 AMjames_bond
This is about an acute triangle so ...