Derive the inequality


by tehno
Tags: derive, inequality
tehno
tehno is offline
#1
Nov23-06, 01:55 PM
P: 363
For a triangle with sides a,b,c and its corresponding circle with radius R:



[tex]\frac{a^2b^2}{c^2} +\frac{a^2c^2}{b^2}+\frac{b^2c^2}{a^2} \geq 9R^2 [/tex]
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Office_Shredder
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#2
Nov23-06, 02:05 PM
Mentor
P: 4,499
What is the corresponding circle? Inscribed or circumscribed?
tehno
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#3
Nov23-06, 02:14 PM
P: 363
Circumscribed of course (usually denoted by capital R).

tehno
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#4
Nov24-06, 11:14 AM
P: 363

Derive the inequality


Anybody?
I prooved it in a long and ugly way.
I'd like to see ,if possible,an elegant proof of it.
murshid_islam
murshid_islam is offline
#5
Nov29-06, 09:04 AM
P: 361
Quote Quote by tehno View Post
Anybody?
I prooved it in a long and ugly way.
I'd like to see ,if possible,an elegant proof of it.
me too.
and tehno, can you please post your proof or at least an outline of it?
tehno
tehno is offline
#6
Dec2-06, 12:30 PM
P: 363
Quote Quote by murshid_islam View Post
me too.
and tehno, can you please post your proof or at least an outline of it?
Nothing particularly smart.
I used substitution:
[tex]R=\frac{abc}{4P}[/tex]
where P is area of triangle with sides a,b,c.
I went to proove :
[tex]\frac{ab}{c^2}+\frac{ac}{b^2}+\frac{bc}{a^2}>\frac{9}{4}[/tex]
which I used along the way to proove the original inequality having on mind basic triangle inequality [tex]a+b>c[/tex].
After lot of algebraic work I arrived at the original inequality.

But can we somehow make a use something more elegant like a well known :
[tex]\frac{1}{a^2+b^2+c^2}\geq \frac{1}{9R^2}[/tex]

?
tehno
tehno is offline
#7
Dec20-06, 10:25 AM
P: 363
Quote Quote by murshid_islam View Post
me too.
and tehno, can you please post your proof or at least an outline of it?
It can't be possible you have a proof of it becouse the inequality is invalid!
Some things were odd and by closer inspection I found error in my proof.
The error helped me also to find an obvious counterexample when ineqality doesn't hold.Consider the triangle with following parameters:
[tex]a=b=1;c=\sqrt{3};R=1[/tex]
tehno
tehno is offline
#8
Jan23-07, 08:18 AM
P: 363
I will rewrite the expression in a trigonometric form and give a restriction.

[tex](sin(\alpha) sin(\beta) cosec(\gamma))^2+(sin(\alpha) cosec(\beta) sin(\gamma))^2+ (cosec(\alpha) sin(\beta) sin(\gamma))^2\geq\frac{9}{4}[/tex]

The restriction is the inequality holds for acute triangles.Now,when I fixed it,
proove the claim.


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