If $A, B, C$ are the $3$ coordinates of the vertices of $\bigtriangleup ABC$, then the area of the triangle is $\frac{1}{2} |\delta_{ABC}|$ where, $$\delta_{ABC}=(x_1-x_2)(y_2-y_3)-(y_1-y_2)(x_2-x_3)$$ We can also write the area without the modulus or the absolute value of $\delta_{ABC}$ but this may result the area to be negative because $\delta_{ABC}$ could be negative depending on the clockwise/anticlockwise loop I'm creating when choosing the order of the vertices.
We can also write our well known area of a triangle formula to be equal to the former formula as (without the modulus): $$\frac{1}{2} \delta_{ABC} = \frac{1}{2} AB \times CN $$ only if we consider $N$ to be the coordinate of the foot of the altitude from vertex $C$ to the opposite side $AB$. Visualize it.
Context:
Look at this diagram. If $CN$ and $DM$ both are perpendicular to the line $AB$, then $\bigtriangleup CNE$ $\bigtriangleup DME$ is simillar. Now, consider by $SSS$ similarity $$\frac{CN}{DM}=\frac{CE}{DE}=\frac{m_1}{m_2}$$ $$\therefore \frac{\bigtriangleup ABC}{\bigtriangleup ABD}\bbox[yellow,5px]{= \frac{\frac{1}{2} \delta_{ABC}}{\frac{1}{2} \delta_{ABD}} = \frac{\frac{1}{2} AB \times CN}{\frac{1}{2} AB \times DM}}=\frac{m_1}{m_2}$$
$\implies \frac{m_1}{m_2}=\frac{ \delta_{ABC}}{\delta_{ABD}}$ the ratio positive $(+)$ and negative $(-)$ implies $AB$ divides $CD$ line segment at the point $E$ externally and internally respectively. Now let's back to my original question which hopefully doesn't require the context.
$\pmb{My~ Question:}$
The question that is troubling me is, if you look at my sample diagram, $AB$ and $CN$ both looks positive to me as they are distance between points and I have this written $\frac{1}{2} \delta_{ABC} = \frac{1}{2} AB \times CN $ when deriving the formula above in coordinate geometry in my textbook but I can't understand if I write $\frac{1}{2} AB \times CN$ instead of $\frac{1}{2} \delta_{ABC}$ (where we might get negative value) then where does scope of being negative comes in as $AB$ and $CN$ both should be positive?