If `G_1` is the mean centre of `A_1,B_1,C_1 and G_2` that of `A_2,B_2,C_2` then show that `vec(A_1A_2)+vec(B_1B_2)+vec(C_1C_2)=3vec(G_1G_2)`

If G is the centroid of Delta ABC and G' is the centroid of Delta A' B' C' " then " vec(A A)' + vec(B B)' + vec(C C)' =

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

Let a ,b be positive real numbers. If a, A_1, A_2, b be are in arithmetic progression a ,G_1, G_2, b are in geometric progression, and a ,H_1, H_2, b are in harmonic progression, show that (G_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)

If the points (a_1, b_1),\ \ (a_2, b_2) and (a_1+a_2,\ b_1+b_2) are collinear, show that a_1b_2=a_2b_1 .

If a is the A.M. of b and c and the two geometric means are G_1 and G_2 , then prove that G1^3+G2^3=2a b c

Let a ,b be positive real numbers. If a A_1, A_2, b be are in arithmetic progression a ,G_1, G_2, b are in geometric progression, and a ,H_1, H_2, b are in harmonic progression, show that (G_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)=((2a+b)(a+2b))/(9a b)

If A_1,A_2,G_1,G_2 and H_1,H_2 are AM’s, GM’s and HM’s between two numbers, then (A_1+A_2)/(H_1+H_2). (H_1H_2)/(G_1G_2) equals _____

Let vec r_1, vec r_2, vec r_3, , vec r_n be the position vectors of points P_1,P_2, P_3 ,P_n relative to the origin Odot If the vector equation a_1 vec r_1+a_2 vec r_2++a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a. a_1+a_2+dot+a_n=n b. a_1+a_2+dot+a_n=1 c. a_1+a_2+dot+a_n=0 d. a_1=a_2=a_3dot+a_n=0

If G is the centroid of a triangle A B C , prove that vec G A+ vec G B+ vec G C= vec0dot

If G is the centroid of a triangle A B C , prove that vec G A+ vec G B+ vec G C= vec0dot