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  • If G is the CENTROID of triangle ABC then prove that AB?2 +BC?2+AC?2=3(GA?2+GB?2+GC?2) BEST OF LUCK FOR THE EQUATION

If G is the CENTROID of triangle ABC then prove that AB?2 +BC?2+AC?2=3(GA?2+GB?2+GC?2) BEST OF LUCK FOR THE EQUATION

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Let A(x_1,y_1),B(x_2,y_2);and;C(x_3,y_3),;be;the;vertices;of; riangle ABC\*Assume;the;centroid;of;the; riangle ABC;to;be;at;the;origin\* Centroid=(0,0)= (fracx_1+x_2+x_33,fracy_1+y_2+y_33)\* Rightarrow x_1+x_2+ x_3=0;and;y_1+y_2+y_3=0\* squaring;both;side,;we;get\*x_1^2+ x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_3x_1=0,;and\*y_1^2+y_2^2+y_3^2+2y_1y_2+2y_2y_3+2y_3y_1=0;;;;;;;..eq(1)\*Rightarrow AB^2+BC^2 +CA^2 \* =[(x_2-x_1)^2+(y_2-y_1)^2]+[(x_3-x_2)^2+(y_3- y_2)^2]+[(x_1-x_3)^2+(y_1-y_3)^2]\* = [(x_1^2+x_2^2-2x_1x_2+y_1^2+ y_2^2-2y_1 y_2)+(x_2^2+x_3^2-2x_2x_3+y_2^2+y_3^2-2y_2y_3)+(x_1^2+x_3^2-2 x_1x_3+y_1^2+y_3^2-2y_1y_3)\*using;..eq(1),\*=3(x_1^2+x_2^2+ x_ 3^ 2)+3(y_1^2+ y_2^2+y_3^2);;;;;..eq(2)\* Rightarrow 3(GA^2 + GB^2 + GC^2)\*=3[(x_1-0)^2+(y_1-0)^2+(x_2-0)^2 +(y_2-0)^2+(x_3-0)^2+(y_3-0)^2]\* =3(x_1^2+x_2^2+x_3^2)+3(y_1^2+ y_2^2+y_3^2);;;;..eq(3)\* from..eq(1);and;..eq(2)\* AB^2+BC^2+CA^2 =3(GA^2+GB^2+GC^2)

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Deependra Verma

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