IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

NITISHA

If G is the centroid of triangle ABC &O is any other point ,prove that

OA²+OB²+OC²=GA²+GB²+GC²+3GO^2.

PLEASE HELP...

spideyunlimited

can u provide a diagram? and is the triangle equilateral or not mentioned. and is point O lying on the triangle? or inside or outside?

NITISHA

nothing else has been mentioned in the question..

nunoxic

Try Using Apolonius Principle
If I Am Not Wrong.......This is a sum from the 2nd exercise of Loney...am i right ?

vishnu_va

We can also prove this by analytical geometry.

Assume the vertices triangle be (x,y);(h,k);(a,b); and take O as any value like (l,m).

Find 0A,0B and OC using distance formula and substitute in LHS.

OR

Find GA,GB,GC and OG and substitute the values in RHS.

You will get the answer.

spideyunlimited

its not so easy that way..im thinking of a method

aankurverma

A(x1,y1),B(x2,y2),C(x3,y3)

n O (h,k) G(sigma x1/3. y1/3)

OA^2 + OB^2 + OC^2 = (h - x1)^2 + (k - y1)^2 + ....

(x1^2 + y1^2) + (x2^2 + y2^2) + (x3^2 + y3^2) + 3(h^2 + k^2) -2h(x1 + x2 + x3) - 2k(y1 + y2 + y3)

GA^2 + GB^2 + gc^2 + 3GO^2 - 0 - 0 {GO^@ = (H - O )^2 + (K - O)^2}

sowjanya_reddygudipati

hey i think this is a 3d question

i have the soln in 3 d

let the vector OA=a and the vector OB=b and the vector OC=c.centroid of the triangle is given by OG=a+b+c/3

then (OA)²+(OB)²+(OC)²=a²+b²+c²

then GA²=((b+c-2a)/3)² ,GB²=((a+c-2b)/3)²,GC²=((b+a-2c)/3)²

then GA²+GB²+GC²+3OG²=9a²+9b²+9c²/9 this is equal to OA²+OB²+OC²