If G b the centroid of a triangle ABC n O b any other point then prove that :
1.) AB2+ BC2+ CA2 ≡ 3( GA2+ GB2+ GC2)
2.)OA 2+ OB 2+ OC 2 = GA 2+ GB 2+ GC 2+ 3GO 2
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2 Answers
2305From the previous question we get,
OA^2+OB^2+OC^2 = GA^2+GB^2+GC^2+3GO^2
ReplacinG O with A,
AB^2+AC^2=GA^2+GB^2+GC^2+3GA^2
Replacing O with B,
AB^2+BC^2=GA^2+GB^2+GC^2+3GB^2
Replacing O with C,
AC^2+BC^2=GA^2+GB^2+GC^2+3GC^2
Adding these equations we get,
AB^2+BC^2+AC^2=3(GA^2+GB^2+GC^2)
Himanshu Giria ThanksUpvote·0· Reply ·2013-11-05 16:33:51
23052)
\vec{G}=\frac{\vec{A}+\vec{B}+\vec{C}}{3}\dots (1)
OA^2+OB^2+OC^2
= (\vec{OG}+\vec{GA})^2+(\vec{OG}+\vec{GB})^2+(\vec{OG}+\vec{OC})^2
= 3GO^2 +GA^2+GB^2+GC^2+2\vec{OG}(\vec{GA}+\vec{GB}+\vec{GC})
\vec{GA}+\vec{GB}+\vec{GC}
=\vec{3G}-\vec{A}-\vec{B}-\vec{C}=0
\therefore OA^2+OB^2+OC^2 = 3GO^2 +GA^2+GB^2+GC^2