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Question

If G be the centroid of a triangle ABC, prove that AB2+BC2+AC2=3GA2+GB2+GC2.


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Solution

Step 1: Find the centroid of the triangle

Given:

  • G is the centroid of a triangle ABC
  • Let G=(0,0) and A=(x1,y1),B=(x2,y2),C=(x3,y3)

Centroid of ABC=x1+x2+x33,y1+y2+y33

(0,0)=x1+x2+x33,y1+y2+y33

Step 2: Find the sides AB,BC,CA,GA,GB,GC

so, x1+x2+x3=0,y1+y2+y3=0

squaring both sides

x1+x2+x32=0,y1+y2+y32=0x12+x22+x32+2x1x2+2x2x3+2x1x3=0-----(1)

and y12+y22+y32+2y1y2+2y2y3+2y1y3=0------(2)

AB=(x2-x1)2+(y2-y1)2,BC=(x3-x2)2+(y3-y2)2,AC=(x3-x1)2+(y3-y1)2

and GA=(x1-0)2+(y1-0)2,GB=(x2-0)2+(y2-0)2,GC=(x3-0)2+(y3-0)2GA=x12+y12,GB=x22+y22,GC=x32+y32GA2+GB2+GC2=x12+x22+x32+y12+y22+y32------(3)

Step 3: Simplify the LHS

AB2+BC2+CA2=x2-x12+y2-y12+x3-x22+y2-y12+(x3-x1)2+(y3-y1)2AB2+BC2+CA2=x12+x22+x32+x12+x22+x32-2x1x2-2x2x3-2x1x3+y12+y22+y32+y12+y22+y32-2y1y2-2y2y3-2y1y3AB2+BC2+CA2=2x12+x22+x32+2y12+y22+y32-2x1x2+2x2x3+2x1x3-2y1y2+2y2y3+2y1y3AB2+BC2+CA2=2x12+x22+x32+2y12+y22+y32+x12+x22+x32+y12+y22+y32usingequation(1)and(2)AB2+BC2+CA2=3x12+x22+x32+3y12+y22+y32AB2+BC2+CA2=3x12+x22+x32+y12+y22+y32AB2+BC2+CA2=3GA2+GB2+GC2usingequation(3)

Hence, AB2+BC2+AC2=3GA2+GB2+GC2 is proved.


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