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#1 2009-08-10 00:41:12


prove it!

If a,b,c r are the lengths of a triangle whose area is S , then show that


#2 2009-08-10 04:41:25

Registered: 2009-07-05
Posts: 59

Re: prove it!


I think I have found the solution based on this website:

Lets go:

According to Heron's formula (

So, we are going to try to maximize the are of the triangle, subject to k = a^2 + b^2 + c^2 being constant, using Lagrange multipliers.

It is easier to work with S^2, and maximize it, so we set:

I felt lazy to solve the system of equations myself, so I plugged it into Mathematica (

The only valid solution with all a, b, c > 0 yields:

Therefore, and since we were trying to maximize S, we have:

Hope it helps! smile

“Make everything as simple as possible, but not simpler.” -- Albert Einstein


#3 2009-08-10 12:25:21

Registered: 2006-07-18
Posts: 280

Re: prove it!

We first use the cosine theorem on each of the squared sides:

Thus we get:

solving for
and using the areaforumla (
etc) :

To minimize this sum we can for example use jensens inequality.
We first assume that the triangle is acute, so all angles are less than
. Then
is convex, since
. Thus Jensens inequality applies and we get:


What if one angle is obtuse?
Then we can argue as follows. WLOG we assume

. Then there exists a corresponding triangle with sides a',b',c' and
and b=b' and c=c'. This triangle must have the same area since
but a'<a since it corresponds to a smaller angle. the new triangle is acute, so we can apply the result there, thus



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