If x+y+z=0 then show that `abs((1,1,1),(x,y,z),(x^3,y^3,z^3))` = 0.

if x+y+z=0 , then show that, |{:(1,1,1),(x,y,z),(x^3,y^3,z^3):}|=0

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

If x + y + z = 0 , then prove that x^(3)+y^(3)+z^(3)= 3xyz

If x/a = y/b = z/c , then show that x^(3)/a^(3) + y^(3)/b^(3) + z^(3)/c^(3) = 3 ((x+y+z)/(a+b+c))^(3) .

If x,y,z are all distinct and if abs((x,x^2,1+x^3),(y,y^2,1+y^3),(z,z^2,1+z^3)) =0,show that xyz+1=0

If z = x + iy and |2z+1|=|z-2i| ,then show that 3(x^(2)+y^(2))+4(x+y)=3

If x+y+z=xyz then prove that (3x-x^3)/(1-3x^2)+(3y-y^3)/(1-3y^2)+(3z-z^3)/(1-3z^2)=(3x-x^3)/(1-3x^2).(3y-y^3)/(1-3y^2).(3z-z^3)/(1-3z^2)

|{:(x,z,0),(0,y,y),(z,0,x):}|

If (x_(1),y_(1),z_(1)) , (x_(2),y_(2),z_(2)) , (x_(3) ,y_(3),z_(3)) and (x_(4) , y_(4) , z_(4)) be the consecutive vertices of a parallelogram, show that x_(1)+x_(3)=x_(2)+x_(4),y_(1)+y_(3)=y_(2)+y_(4) and z_(1)+z_(3)=z_(2)+z_(4) .