# Coordinate geometry - centroid (SL LONEY exercise problem)

1. Mar 4, 2017

### matrixone

1. The problem statement, all variables and given/known data

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

2. Relevant 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...

3. 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 cant even get a start .....

2. Mar 4, 2017

### 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')?

3. Mar 4, 2017

### 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 ?

4. Mar 4, 2017

### haruspex

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?

5. Mar 4, 2017

### matrixone

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

Last edited: Mar 4, 2017
6. Mar 4, 2017

### haruspex

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.

7. Mar 4, 2017

### Buffu

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.

Last edited: Mar 4, 2017
8. Mar 4, 2017

### Buffu

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

9. Mar 7, 2017

### matrixone

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

I cant 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 cant see were this leads to the actual problem ....Or, were you thinking in some other way ?

10. Mar 10, 2017

### issacnewton

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

11. Mar 10, 2017

### haruspex

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

12. Mar 10, 2017

### Buffu

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

13. Mar 10, 2017

### haruspex

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.

14. Mar 10, 2017

### issacnewton

Buffu, what primitive method you are talking ?