Choose whether the given statement is true or false.

If G be the centroid of a triangle ABC and O be any other point, is it possible to prove that $3 (G A^{2} + G B^{2} + G C^{2}) = B C^{2} + C A^{2} + A B^{2}$ and $O A^{2} + O B^{2} + O C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 G O^{2}$ ?

Question

Choose whether the given statement is true or false.

If G be the centroid of a triangle ABC and O be any other point, is it possible to prove that $3 (G A^{2} + G B^{2} + G C^{2}) = B C^{2} + C A^{2} + A B^{2}$ and $O A^{2} + O B^{2} + O C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 G O^{2}$ ?

Accepted Answer

According to the problem, G be the centroid of a triangle ABC and O be any other point. We need to prove 3GA2+GB2+GC2=BC2+CA2+AB2 and OA2+OB2+OC2= GA2+GB2+GC2+3GO2.Let us assume the vertices A, B and C be (x1,y1), (x2,y2), (x3,y3),We know that the centroid G of a triangle is defined as A+B+C3.So, we get G=(x1+x2+x33, y1+y2+y33)We know that the distance between two points (a,b) and (c,d) is defined as(a-c)2+(b-d)2 .So, we get AB2=(x2-x12+y2-y12)2⇒AB2=x12+x12-2x1x2+y12+y22-2y1y2⇒AB2=(x12+x12)+(y12+y22)-2(x1x2+y1y2)Similarly, we get BC2=(x22+x32)+( (y22+y32)−2(x2x3+y2y3)and CA2=(x32+x12)+( y32+y12)−2(x3x1+y3y1).Let us consider AB2+ BC2+CA2.⇒AB2+ BC2+CA2=2(x12+x22+x32)+2(y12+y22+y32)−2(x1x2+x2x3+x3x1+y1y2+y2y3+y3y1)---(1).Now, let us find GA2,GB2 and GC2.So, we get GA2=(x1-x1+x2+x33)2+(y1-y1+y2+y33)2⇒GA2=(2x1-x2-x33)2+(2y1-y2-y33)2.⇒ GA2=(4x12+x22+x32-4x1x2+2x2x3-4x3x19)2+(4y12+y22+y32-4y1y2-2y2y3-4y3y19)2⇒ GA2=(4(x12+y12)+(x22+y22)+(x32+y32)-4(x1x2+y1y2)+2(x2x3+y2y3)-4(x3x1+y1y3)9).Similarly, we get ⇒GB2=((x12+y12)+4(x22+y22)+(x32+y32)-4(x1x2+y1y2)-4(x2x3+y2y3)+2(x3x1+y1y3)9).,⇒ GC2=((x12+y12)+4(x22+y22)+4(x32+y32)+2(x1x2+y1y2)-4x2x3+y2y3-4(x3x1+y1y3)9).Now, let us consider GA2+GB2+GC2 .GA2+GB2+GC2=(6(x12+y12)+6(x22+y22)+6(x32+y32)-6(x1x2+y1y2)-6x2x3+y2y3-6(x3x1+y1y3)9).GA2+GB2+GC2=(2(x12+y12)+2(x22+y22)+2(x32+y32)-2(x1x2+y1y2)-2x2x3+y2y3-2(x3x1+y1y3)3)….(2).From equation (1), we get 3(GA2+GB2+GC2)= AB2+ BC2+CA2So, we have proved 3(GA2+GB2+GC2)= AB2+ BC2+CA2---(3).Now, let us assume the point O be (x,y).Now, let us find  OA2 .So, we get OA2=(x1-x2+y1-y2)2⇒OA2=x12+x2-2xx1+y12+y2-2yy1⇒OA2=x12+y12+(x2+y2)-2(xx1+yy1)Similarly, we get OB2=x22+y22+(x2+y2)- 2(xx2+yy2) and OA2=x32+y32+(x2+y2)- 2(xx3+yy3)Let us consider OA2+OB2+OC2.⇒OA2+OB2+OC2=(x12+x22+x32+y12+y22+y32)+3(x2+y2)-2(xx1+xx2+xx3+yy1+yy2+yy3)⇒OA2+OB2+OC2=(x12+x22+x32+y12+y22+y32)+3(x2+y2)-2x(x1+x2+x3)-2y(y1+y2+y3)---(4).Now, let us find GO2⇒GO2=(x-x1+x2+x332+y-y1+y2+y332)2⇒GO2=x2+x1+x2+x332-2 xx1+x2+x33+y2+y1+y2+y332-2yy1+y2+y33 ⇒ GO2=x2+y2+x12+x22+x32+y12+y22+y32+2x1x2+2x2x3+2x3x1+2y1y2+2y2y3+2y1y39-(2xx1+x2+x3+2y(y1+y2+y3)3).⇒3GO2=3(x2+y2) +x12+x22+x32+y12+y22+y32+2x1x2+2x2x3+2x3x1+2y1y2+2y2y3+2y1y33-2xx1+x2+x3-2y(y1+y2+y3) .Using equation (2), we getGA2+GB2+GC2+3GO2=3x2+y2+3(x12+y12)+3(x22+y22)+3(x32+y32)3-2xx1+x2+x3-2y(y1+y2+y3) ⇒GA2+GB2+GC2+3GO2=3x2+y2+x12+y12+x22+y22+x32+y32-2xx1+x2+x3-2y(y1+y2+y3) From equation (4), we getOA2+OB2+OC2=GA2+GB2+GC2+3GO2 ---(5).From equations (3) and (5), we have proved 3(GA2+GB2+GC2)= BC2+CA2+AB2 and⇒OA2+OB2+OC2=GA2+GB2+GC2+3GO2

Choose whether the given statement is true or false.

If G be the centroid of a triangle ABC and O be any other point, is it possible to prove that $3 (G A^{2} + G B^{2} + G C^{2}) = B C^{2} + C A^{2} + A B^{2}$ and $O A^{2} + O B^{2} + O C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 G O^{2}$ ?

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App

Choose whether the given statement is true or false.If G be the centroid of a triangle ABC and O be any other point, is it possible to prove that 3GA2+GB2+GC2=BC2+CA2+AB2 and OA2+OB2+OC2= GA2+GB2+GC2+3GO2?

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App

Choose whether the given statement is true or false.

If G be the centroid of a triangle ABC and O be any other point, is it possible to prove that $3 (G A^{2} + G B^{2} + G C^{2}) = B C^{2} + C A^{2} + A B^{2}$ and $O A^{2} + O B^{2} + O C^{2} = G A^{2} + G B^{2} + G C^{2} + 3 G O^{2}$ ?