Let `z_1, z_2, z_3` be three complex numbers and `a ,b ,c` be real numbers not all zero, such that `a+b+c=0a n da z_1+b z_2+c z_3=0.` Show that `z_1, z_2,z_3` are collinear.

If z_1, z_2, z_3 are three complex numbers such that 5z_1-13 z_2+8z_3=0, then prove that [(z_1,(bar z )_1, 1),(z_2,(bar z )_2 ,1),(z_3,(bar z )_3 ,1)]=0

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot

If z_(1),z_(2), and z_(3) are three complex numbers such that |z_(1)| = 1, " " |z_(2)| = 2, " " |z_(3)| = 3 and |z_(1) + z_(2) + z_(3) | = 1, show that |9z_(1)z_(2) + 4 z_(1)z_(3) + z_(2)z_(3)| = 6 .

If z_1, z_2, z_3 are distinct nonzero complex numbers and a ,b , c in R^+ such that a/(|z_1-z_2|)=b/(|z_2-z_3|)=c/(|z_3-z_1|) Then find the value of (a^2)/(z_1-z_2)+(b^2)/(z_2-z_3)+(c^2)/(z_3-z_1)

If z_1, z_2, z_3 be the affixes of the vertices A, B Mand C of a triangle having centroid at G such ;that z = 0 is the mid point of AG then 4z_1 + Z_2 + Z_3 =

z_(1), z_(3), and z_(3) are complex numbers such that z_(1) + z_(2) + z_(3)=0 and |z_(1)| = |z_(2)| = |z_(3)| = 1 then z_(1)^(2)+z_(2)^(2)+z_(3)^(3)

If z_1, z_2, z_3 are three nonzero complex numbers such that z_3=(1-lambda)z_1+lambdaz_2w h e r elambda in R-{0}, then prove that points corresponding to z_1, z_2a n dz_3 are collinear .

Let z_1, z_2a n dz_3 represent the vertices A ,B ,a n dC of the triangle A B C , respectively, in the Argand plane, such that |z_1|=|z_2|=|z_3|=5. Prove that z_1sin2A+z_2sin2B+z_3sin2C=0.