Coordinate geometry - centroid (SL LONEY exercise problem)

[matrixone]

Homework Statement

If G be the centroid of ΔABC and O be any other point, prove that ,
$3(GA^2 + GB^2 + GC^2)=BC^2+CA^2+AB^2$
$and,$
$OA^2 + OB^2 + OC^2 = GA^2.GB^2+GC^2+3GO^2$

Homework Equations

i m practising from S L LONEY coordinate geometry first chapter ... only the equation that i used to solve is mentioned in the chapter...

The Attempt at a Solution

i felt that co ordinate geometry approach will be cumbersome ...
so i started with
$GA^2 + GB^2 = AB^2+2.GA.GB.cos∠AGB$
and similar 2 more equations
But i am stuck here ! :(

And for the 2nd question i can't even get a start ...

Please help me with a hint ...
[/B]

 

[haruspex]

If you have a triangle ABC and drop a median from A to A', the midpoint of BC, what is the relationship between the lengths AB, AC, AA' and BA'(=CA')?

 

[matrixone]

I can't find any sir... : ( ...definitely , ΔABA' and ΔACA' are not similar ...
are you referring to the same cosine rule that i used ?

 

[haruspex]

I can't find any sir... : ( ...definitely , ΔABA' and ΔACA' are not similar ...
are you referring to the same cosine rule that i used ?

No. It is a result I dimly recall from my own schooldays, so I thought you may have encountered it in your book.
Drop a perpendicular from A to BC meeting it at P. Call AP length h and PA' length x. What three equations can you write relating these to AB, AC etc. using Pythagoras' Theorem?

 

[matrixone]

No. It is a result I dimly recall from my own schooldays, so I thought you may have encountered it in your book.
Drop a perpendicular from A to BC meeting it at P. Call AP length h and PA' length x. What three equations can you write relating these to AB, AC etc. using Pythagoras' Theorem?

BP² + h² = AB²
PC² + h² = AC²
x² + h² = AA'²

 

[haruspex]

BP² + h² = AB²
PC² + h² = AC²
x² + h² = AA'²

There's a mistake in the first one, probably just typed wrongly.
You need to get rid of PC. What is the relationship between PC, x and A'C?
Next manipulate the equations to eliminate x² and h².

 

[Buffu]

i felt that co ordinate geometry approach will be cumbersome ...
so i started with
GA2+GB2=AB2+2.GA.GB.cos∠AGBGA2+GB2=AB2+2.GA.GB.cos∠AGBGA^2 + GB^2 = AB^2+2.GA.GB.cos∠AGB
and similar 2 more equations
But i am stuck here ! :(

And for the 2nd question i can't even get a start ...

Please help me with a hint ...

Pick a triangle of verticies $(0,0), (a,0), (x,y)$ then $G = \left({a + x \over 3}, {y\over 3}\right)$. Now you evaluate LHS of first equation and then RHS. It is not that cumbersome.

For the second draw the figure and use cosine rule.

 

[Buffu]

Use cosine rule on the vertex G of the red triangle. I am not sure it will work though.

 

[matrixone]

Pick a triangle of verticies $(0,0), (a,0), (x,y)$ then $G = \left({a + x \over 3}, {y\over 3}\right)$. Now you evaluate LHS of first equation and then RHS. It is not that cumbersome.

For the second draw the figure and use cosine rule.

that was optimistic choice of vertices. I solved it that way (by comparing coefficients) and yes, not too cumbersome. thanks :)

There's a mistake in the first one, probably just typed wrongly.
You need to get rid of PC. What is the relationship between PC, x and A'C?
Next manipulate the equations to eliminate x² and h².

I can't see any mistake in the first one sir.. : ( ...and by changing PC = A'C - x , from last two equations i am able to get an expression for x (a big one) and then using that value to get expression for h seems even a time consuming approach .(Buffu's solution seems better)... And i can't see were this leads to the actual problem ...Or, were you thinking in some other way ?

 

[issacnewton]

matrixone, how did you do the second one ? I tried but am stuck on cosines.

 

[haruspex]

I can't see any mistake in the first one sir

Sorry, I thought I replied to this... No, you are right, I misread something.

 

[Buffu]

matrixone, how did you do the second one ? I tried but am stuck on cosines.

I can solve this but my method is too primitive to do.

 

[haruspex]

from last two equations i am able to get an expression for x (a big one

No, you should get quite a simple relationship between AB, AC, A'B and AA'. Please post your working.
It is a well-known theorem.

Once you have that, the original problem is quite easy.

 

[issacnewton]

Buffu, what primitive method you are talking ?