Area of triangle in complex number form

 

The form

        Given that  z1, z2, z3  be the vertices of a triangle, then the area of the triangle is given by:

 

              

 

        where the entries of the third row denote the conjugates of the corresponding complex numbers in the second row.

 

 

Small warm-up exercise

 

If  z1 = x1 + y1i,  z2 = x2 + y2i,  z = x + yi,

Then

 

(1)       ,where I(z) stands for the imaginary part of z.

(2)    

 

 

Proof 1

 

                , by (1)

        ,        , by (2)

 

       

 

       

 

 

 

Proof 2

 

Using the area formula in Cartesian Cordinates: