What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? 
Is it 
$$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$
where $w^*$ denotes the complex conjugate?

Let $z_j = x_j + iy_j$, $j = 1, 2, 3$. The area of the triangle is given by 
\begin{align*}
\frac{1}{2} \begin{vmatrix}
1 & x_1 & y_1 \\
1 & x_2 & y_2 \\
1 & x_3 & y_3 
\end{vmatrix}&= \frac{1}{2} \begin{vmatrix}
1 & x_1+iy_1 & y_1 \\
1 & x_2+iy_2 & y_2 \\
1 & x_3+iy_3 & y_3 
\end{vmatrix}\\
&= \frac{1}{4i} \begin{vmatrix}
1 & z_1 & z_1-z_1^*\\
1 & z_2 & z_2-z_2^* \\
1 & z_3 & z_3-z_3^* 
\end{vmatrix}\\
\end{align*}
Now expand via first column to get the required expression.
