Register to reply 
Derive the inequality 
Share this thread: 
#1
Nov2306, 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] 


#2
Nov2306, 02:05 PM

Emeritus
Sci Advisor
PF Gold
P: 4,500

What is the corresponding circle? Inscribed or circumscribed?



#3
Nov2306, 02:14 PM

P: 363

Circumscribed of course (usually denoted by capital R).



#4
Nov2406, 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. 


#5
Nov2906, 09:04 AM

P: 361

and tehno, can you please post your proof or at least an outline of it? 


#6
Dec206, 12:30 PM

P: 363

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] ? 


#7
Dec2006, 10:25 AM

P: 363

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] 


#8
Jan2307, 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. 


Register to reply 
Related Discussions  
Derive 6  Calculus & Beyond Homework  2  
Another derive  Calculus & Beyond Homework  1  
Proof this inequality using Chebyshev's sum inequality  Calculus & Beyond Homework  1  
How would i derive the equation?  Introductory Physics Homework  3  
Derive an Expression  Introductory Physics Homework  1 