Another proof

**Nichelle14** · July 5th 2006, 04:04 PM

I have to Prove (a+b)^2 + (b+c)^2 + (c+a)^2 >= 4(ab + bc + ac)

I multiplied out both sides and then move everything to the left, but I don't know what I should do after that.

**malaygoel** · July 5th 2006, 06:32 PM

Originally Posted by Nichelle14

I have to Prove (a+b)^2 + (b+c)^2 + (c+a)^2 >= 4(ab + bc + ac)

I multiplied out both sides and then move everything to the left, but I don't know what I should do after that.

$(a+b)^2 + (b+c)^2 + (c+a)^2$
$=2(a^2 + b^2 + c^2)+ 2(ab + bc + ca)$
$=(a^2 + b^2)+(b^2 + c^2)+(a^2 + c^2)+ 2(ab + bc + ca)$
$\geq4(ab + bc + ca)$

Keep Smiling
Malay

**ThePerfectHacker** · July 5th 2006, 06:54 PM

Originally Posted by Nichelle14

I have to Prove (a+b)^2 + (b+c)^2 + (c+a)^2 >= 4(ab + bc + ac)

I multiplied out both sides and then move everything to the left, but I don't know what I should do after that.

It is necessary and suffienct for,
$a^2+b^2+c^2\geq ab+ac+bc$
----
If you play your cards right this is the "Cauchy-Swarchtz Inequality".
Let,
$\bold{v}=(a,b,c)$
$\bold{u}=(b,c,a)$
Then,
$||\bold{u}||\cdot ||\bold{v}||\geq |\bold{u}\cdot \bold{v} |$
Now,
$<br /> ||\bold{u}||\cdot ||\bold{v}||=a^2+b^2+c^2$
And,
$|\bold{u}\cdot \bold{v} |=|ab+bc+ac|$
Thus,
$a^2+b^2+c^2\geq |ab+bc+ac|\geq ab+bc+ac$

**Soroban** · July 6th 2006, 09:18 AM

Hello, Nichelle14!

Prove that: . $(a+b)^2 + (b+c)^2 + (c+a)^2\:\geq\:4(ab + bc + ac)$

I multiplied out both sides and then move everything to the left. . . . Not recommended

Multiply out the left side and we have:
. . $(a+b)^2 + (b+c)^2 + (c+a)^2\:=$ $\:2(a^2 + b^2+c^2) + 4(ab + bc + ac)$ LHS

Since $a,\,b,\,c$ are real numbers: . $a^2 + b^2 + c^2 \:\geq \:0$

Multiply by $2:\;\;2(a^2 + b^2 + c^2) \:\geq \:0$

Add $4(ab + bc + ac)$ to both sides:

. . $\underbrace{2(a^2 + b^2 + c^2) + 4(ab + bc + ac)} \:\geq\:4(ab + bc + ac)$
. . . . . . . . This is LHS

Therefore: . $(a+b)^2 + (b+c)^2 + (c+a)^2\;\geq\;4(ab + bc + ac)$

Thread: Another proof

LinkBack

Thread Tools

Display

Another proof

Similar Math Help Forum Discussions

Proof theory: example of an existence proof of a proof?

Convert the following equational proof into a Hilbert Proof

[SOLVED] direct proof and proof by contradiction

Proof with algebra, and proof by induction (problems)

proof that the proof that .999_ = 1 is not a proof (version)

Search Tags