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 ).
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The area of the triangle.
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.
Originally Posted by proscientia By Heron’s formula for the area of the triangle, 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.
Yes, I’ve just noticed that. It was very careless of me. Sorry.
Last edited by proscientia; Oct 28th 2009 at 02:09 PM.
Hmm, I wondering if what is to be proved is actually false. Heronís formula gives (hopefully thatís correct this time). Then However it is possible that this inequality may not be true for arbitrarily close to See this.
This is about an acute triangle so ...
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