For any complex number z show that `zbar z=|z|^2`

If z_1 , and z_2 be two complex numbers prove that z_1barz_1=|z_1|^2

Write any two complex numbers,then show that |z1+z2|^2+|z1-z2|^2=2(|z1|^2+|z2|^2)

If z1=x1+iy1 and z2=x2+iy2 be two complex number,then show that bar(z1,z2)=bar z1.bar z2 .

For any two complex numbers z1 and z2 ,prove that |z1.z2|=|z1|.|z2|

For any two complex numbers z1 and z2 ,prove that arg(z1.z2)=arg(z1)+arg(z2) .

For all two complex numbers z1 and z2,prove that Re(z1z2)=Re z1 Re z2-Im z1 Im z2

If z_1 , and z_2 be two complex numbers prove that bar(z_1+-z_2)=barz_1+-barz_2

If z be a complex number, show that the minimum value of |z| + |z - 1| is 1.

If z_1 , and z_2 be two complex numbers prove that |z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]

If z_1 , z_2 , z_3 represent vertices of an equilateral triangle such that |z_1| = |z_2| = |z_3| show that z_1+z_2+z_3=0