Question

Show that the triangle with vertices at the points $z_1,z_2$ and $(1-i)z_1+{iz}_2$ is right angled and isosceles.

Answer 1

If the given points be $A,B,C$
$\left|AB\right|=|z_1-z_2|$,
$\left|BC\right|=|(1-i)z_1-(1-i)z_2|$
$=|1-i||z_1-z_2|=\sqrt{2}|z_1-z_2|$
$\left|CA\right|=\left|i(z_1-z_2)\right|=\left|i\right||z_1-z_2|=|z_1-z_2|$
Clearly $AB=CA$ and also
${AB}^2+{CA}^2={BC}^2=2{|z_1-z_2|}^2$
$\because BC=\sqrt{2}|z_1-z_2|$
Hence the triangle is right angled isosceles.