If `(b_2-b_1)(b_3-b-1)+(a_2-a_1)(a_3-a_1)=0`, then prove that the circumcenter of the triangle having vertices `(a_1,b_1),(a_2,b_2)` and `(a_3,b_3)` is `((a_2+a_3)/(2),(b_2+b_3)/(2))`.

Find the coordinates of the centroid of the triangle with its vertices at (a_1,b_1,c_1),(a_2,b_2,c_2), "and"(a_3, b_3,c_3).

If the line a_1x+b_1y=1,a_2x+b_2y=1,a_3x+b_3y=1 are concurrent then the points (a_1,b_1),(a_2,a_2),(a_3,b_3) ,

Prove by vector method that (a_1b_1+a_2b_2+a_3b_3)^2lt+(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

Prove that the value of each the following determinants is zero: |[a_1,l a_1+mb_1,b_1],[a_2,l a_2+mb_2,b_2],[a_3,l a_3+m b_3,b_3]|

The determinant |(b_1+c_1,c_1+a_1,a_1+b_1),(b_2+c_2,c_2+a_2,a_2+b_2),(b_3+c_3,c_3+a_3,a_3+b_3)|=

The equation A/(x-a_1)+A_2/(x-a_2)+A_3/(x-a_3)=0 ,where A_1,A_2,A_3gt0 and a_1lta_2lta_3 has two real roots lying in the invervals. (A) (a_1,a_2) and (a_2,a_3) (B) (-oo,a_1) and (a_3,oo) (C) (A_1,A_3) and (A_2,A_3) (D) none of these