2008 Independent Ext2 last question (2 Viewers)

[asianese]

ABC is a triangle with sides a, b, c. If , show that ABC is equilateral.

I have tried most things like cosine rule or sine rule...Then I remembered...

(Which I proved)

And equality occurs iff a=b=c. Hence as , then a=b=c. That is, ABC is equilateral.




Is that valid??

 

[math man]

yeh i think that would be fine, a clever approach, here is cos rule method:

 

[Carrotsticks]

ABC is a triangle with sides a, b, c. If , show that ABC is equilateral.

I have tried most things like cosine rule or sine rule...Then I remembered...

(Which I proved)

And equality occurs iff a=b=c. Hence as , then a=b=c. That is, ABC is equilateral.




Is that valid??

Looks good.

 

[asianese]

I originally got to your 3rd last line and I didn't know how to proceed.

What happened there? Did you just divide by the ab, bc, ca??

Thanks

 

[math man]

cause a,b,c >0 i equated coef's with LHS and RHS and got each as 0, or you can equate from 4th last line

 

[Fus Ro Dah]

ABC is a triangle with sides a, b, c. If , show that ABC is equilateral.

I have tried most things like cosine rule or sine rule...Then I remembered...

(Which I proved)

And equality occurs iff a=b=c. Hence as , then a=b=c. That is, ABC is equilateral.




Is that valid??

I like this method better. Less brute in nature.

 

[math man]

^ true, but it requires to memorise an inequality which you have to prove and im not
a fan of memorising

 

[Fus Ro Dah]

^ true, but it requires to memorise an inequality which you have to prove and im not
a fan of memorising

I'm seeing two contradictory statements. If you have to prove it, then no memorising is required. Also, you used the Cosine Rule. If anything, that is more 'memorising' than asianese's inequality.

 

[math man]

no, cosine rule is not memorsing as we all have used that formula 200000... times, so it is apart of us, whereas
that inequality i doubt we ever use

 

[D94]

^ true, but it requires to memorise an inequality which you have to prove and im not
a fan of memorising

You'd only be "memorising" (a-b)² >= 0, which is one of the classic inequality equations in maths.

 

[RealiseNothing]

You'd only be "memorising" (a-b)² >= 0, which is one of the classic inequality equations in maths.

That's not exactly memorising, it's quite logical and trivial.

 

[Fus Ro Dah]

no, cosine rule is not memorsing as we all have used that formula 200000... times, so it is apart of us, whereas
that inequality i doubt we ever use

But it is probably one of the first inequalities you are taught to prove in Harder Extension 1.

Either way, I like asianese's method more because he proved both directions of the statement, having the 'iff', whereas yours only proves one.

 

[D94]

That's not exactly memorising, it's quite logical and trivial.

Hence why I put the word in quotation marks, emphasising the lack of. You'd be applying that triviality 3 times to result in the inequality asianese used.

 

[math man]

But it is probably one of the first inequalities you are taught to prove in Harder Extension 1.

Either way, I like asianese's method more because he proved both directions of the statement, having the 'iff', whereas yours only proves one.

the question only asked for one direction, so i didnt need to prove the other, but i said from the start his method is better

 

[asianese]

Just to clarify the 'directional' thing is about???

lol seems I opened some pandora's box.

 

[seanieg89]

http://en.wikipedia.org/wiki/Converse_(logic)#Converse_of_a_theorem