Complex numbers and formula for the height

Question

It's a simple question: if $A$, $B$ $C\in\mathbb{C}$ form a triangle on the complex plane, then the formula for the foot (the point of intersection of the altitude dropped from the point $A$ with the base $BC$) of the height $AD$ would be...?

Answer 1

Exactly the same as for $\Bbb{R}^2$ but when you use the $x $ coordinates, you take the real parts, when you use the $y $ coordinates, you use the imaginary parts.

Remember that for $\Bbb {R}^2$ you can have:

If $P $ is the foot, then

$$\begin {cases} AP\cdot BP = 0\\ BP = \lambda BC\end {cases} $$

Making use of the fact that $BP $ is collinear with $BC $ and perpendicular to $AP $.